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Introduction to Differential Equations

Slide Duration:

Section 1: Limits and Derivatives
Overview & Slopes of Curves

42m 8s

Intro
0:00
Overview & Slopes of Curves
0:21
Differential and Integral
0:22
Fundamental Theorem of Calculus
6:36
Differentiation or Taking the Derivative
14:24
What Does the Derivative Mean and How do We Find it?
15:18
Example: f'(x)
19:24
Example: f(x) = sin (x)
29:16
General Procedure for Finding the Derivative of f(x)
37:33
More on Slopes of Curves

50m 53s

Intro
0:00
Slope of the Secant Line along a Curve
0:12
Slope of the Tangent Line to f(x) at a Particlar Point
0:13
Slope of the Secant Line along a Curve
2:59
Instantaneous Slope
6:51
Instantaneous Slope
6:52
Example: Distance, Time, Velocity
13:32
Instantaneous Slope and Average Slope
25:42
Slope & Rate of Change
29:55
Slope & Rate of Change
29:56
Example: Slope = 2
33:16
Example: Slope = 4/3
34:32
Example: Slope = 4 (m/s)
39:12
Example: Density = Mass / Volume
40:33
Average Slope, Average Rate of Change, Instantaneous Slope, and Instantaneous Rate of Change
47:46
Example Problems for Slopes of Curves

59m 12s

Intro
0:00
Example I: Water Tank
0:13
Part A: Which is the Independent Variable and Which is the Dependent?
2:00
Part B: Average Slope
3:18
Part C: Express These Slopes as Rates-of-Change
9:28
Part D: Instantaneous Slope
14:54
Example II: y = √(x-3)
28:26
Part A: Calculate the Slope of the Secant Line
30:39
Part B: Instantaneous Slope
41:26
Part C: Equation for the Tangent Line
43:59
Example III: Object in the Air
49:37
Part A: Average Velocity
50:37
Part B: Instantaneous Velocity
55:30
Desmos Tutorial

18m 43s

Intro
0:00
Desmos Tutorial
1:42
Desmos Tutorial
1:43
Things You Must Learn To Do on Your Particular Calculator
2:39
Things You Must Learn To Do on Your Particular Calculator
2:40
Example I: y=sin x
4:54
Example II: y=x³ and y = d/(dx) (x³)
9:22
Example III: y = x² {-5 <= x <= 0} and y = cos x {0 < x < 6}
13:15
The Limit of a Function

51m 53s

Intro
0:00
The Limit of a Function
0:14
The Limit of a Function
0:15
Graph: Limit of a Function
12:24
Table of Values
16:02
lim x→a f(x) Does not Say What Happens When x = a
20:05
Example I: f(x) = x²
24:34
Example II: f(x) = 7
27:05
Example III: f(x) = 4.5
30:33
Example IV: f(x) = 1/x
34:03
Example V: f(x) = 1/x²
36:43
The Limit of a Function, Cont.
38:16
Infinity and Negative Infinity
38:17
Does Not Exist
42:45
Summary
46:48
Example Problems for the Limit of a Function

24m 43s

Intro
0:00
Example I: Explain in Words What the Following Symbols Mean
0:10
Example II: Find the Following Limit
5:21
Example III: Use the Graph to Find the Following Limits
7:35
Example IV: Use the Graph to Find the Following Limits
11:48
Example V: Sketch the Graph of a Function that Satisfies the Following Properties
15:25
Example VI: Find the Following Limit
18:44
Example VII: Find the Following Limit
20:06
Calculating Limits Mathematically

53m 48s

Intro
0:00
Plug-in Procedure
0:09
Plug-in Procedure
0:10
Limit Laws
9:14
Limit Law 1
10:05
Limit Law 2
10:54
Limit Law 3
11:28
Limit Law 4
11:54
Limit Law 5
12:24
Limit Law 6
13:14
Limit Law 7
14:38
Plug-in Procedure, Cont.
16:35
Plug-in Procedure, Cont.
16:36
Example I: Calculating Limits Mathematically
20:50
Example II: Calculating Limits Mathematically
27:37
Example III: Calculating Limits Mathematically
31:42
Example IV: Calculating Limits Mathematically
35:36
Example V: Calculating Limits Mathematically
40:58
Limits Theorem
44:45
Limits Theorem 1
44:46
Limits Theorem 2: Squeeze Theorem
46:34
Example VI: Calculating Limits Mathematically
49:26
Example Problems for Calculating Limits Mathematically

21m 22s

Intro
0:00
Example I: Evaluate the Following Limit by Showing Each Application of a Limit Law
0:16
Example II: Evaluate the Following Limit
1:51
Example III: Evaluate the Following Limit
3:36
Example IV: Evaluate the Following Limit
8:56
Example V: Evaluate the Following Limit
11:19
Example VI: Calculating Limits Mathematically
13:19
Example VII: Calculating Limits Mathematically
14:59
Calculating Limits as x Goes to Infinity

50m 1s

Intro
0:00
Limit as x Goes to Infinity
0:14
Limit as x Goes to Infinity
0:15
Let's Look at f(x) = 1 / (x-3)
1:04
Summary
9:34
Example I: Calculating Limits as x Goes to Infinity
12:16
Example II: Calculating Limits as x Goes to Infinity
21:22
Example III: Calculating Limits as x Goes to Infinity
24:10
Example IV: Calculating Limits as x Goes to Infinity
36:00
Example Problems for Limits at Infinity

36m 31s

Intro
0:00
Example I: Calculating Limits as x Goes to Infinity
0:14
Example II: Calculating Limits as x Goes to Infinity
3:27
Example III: Calculating Limits as x Goes to Infinity
8:11
Example IV: Calculating Limits as x Goes to Infinity
14:20
Example V: Calculating Limits as x Goes to Infinity
20:07
Example VI: Calculating Limits as x Goes to Infinity
23:36
Continuity

53m

Intro
0:00
Definition of Continuity
0:08
Definition of Continuity
0:09
Example: Not Continuous
3:52
Example: Continuous
4:58
Example: Not Continuous
5:52
Procedure for Finding Continuity
9:45
Law of Continuity
13:44
Law of Continuity
13:45
Example I: Determining Continuity on a Graph
15:55
Example II: Show Continuity & Determine the Interval Over Which the Function is Continuous
17:57
Example III: Is the Following Function Continuous at the Given Point?
22:42
Theorem for Composite Functions
25:28
Theorem for Composite Functions
25:29
Example IV: Is cos(x³ + ln x) Continuous at x=π/2?
27:00
Example V: What Value of A Will make the Following Function Continuous at Every Point of Its Domain?
34:04
Types of Discontinuity
39:18
Removable Discontinuity
39:33
Jump Discontinuity
40:06
Infinite Discontinuity
40:32
Intermediate Value Theorem
40:58
Intermediate Value Theorem: Hypothesis & Conclusion
40:59
Intermediate Value Theorem: Graphically
43:40
Example VI: Prove That the Following Function Has at Least One Real Root in the Interval [4,6]
47:46
Derivative I

40m 2s

Intro
0:00
Derivative
0:09
Derivative
0:10
Example I: Find the Derivative of f(x)=x³
2:20
Notations for the Derivative
7:32
Notations for the Derivative
7:33
Derivative & Rate of Change
11:14
Recall the Rate of Change
11:15
Instantaneous Rate of Change
17:04
Graphing f(x) and f'(x)
19:10
Example II: Find the Derivative of x⁴ - x²
24:00
Example III: Find the Derivative of f(x)=√x
30:51
Derivatives II

53m 45s

Intro
0:00
Example I: Find the Derivative of (2+x)/(3-x)
0:18
Derivatives II
9:02
f(x) is Differentiable if f'(x) Exists
9:03
Recall: For a Limit to Exist, Both Left Hand and Right Hand Limits Must Equal to Each Other
17:19
Geometrically: Differentiability Means the Graph is Smooth
18:44
Example II: Show Analytically that f(x) = |x| is Nor Differentiable at x=0
20:53
Example II: For x > 0
23:53
Example II: For x < 0
25:36
Example II: What is f(0) and What is the lim |x| as x→0?
30:46
Differentiability & Continuity
34:22
Differentiability & Continuity
34:23
How Can a Function Not be Differentiable at a Point?
39:38
How Can a Function Not be Differentiable at a Point?
39:39
Higher Derivatives
41:58
Higher Derivatives
41:59
Derivative Operator
45:12
Example III: Find (dy)/(dx) & (d²y)/(dx²) for y = x³
49:29
More Example Problems for The Derivative

31m 38s

Intro
0:00
Example I: Sketch f'(x)
0:10
Example II: Sketch f'(x)
2:14
Example III: Find the Derivative of the Following Function sing the Definition
3:49
Example IV: Determine f, f', and f'' on a Graph
12:43
Example V: Find an Equation for the Tangent Line to the Graph of the Following Function at the Given x-value
13:40
Example VI: Distance vs. Time
20:15
Example VII: Displacement, Velocity, and Acceleration
23:56
Example VIII: Graph the Displacement Function
28:20
Section 2: Differentiation
Differentiation of Polynomials & Exponential Functions

47m 35s

Intro
0:00
Differentiation of Polynomials & Exponential Functions
0:15
Derivative of a Function
0:16
Derivative of a Constant
2:35
Power Rule
3:08
If C is a Constant
4:19
Sum Rule
5:22
Exponential Functions
6:26
Example I: Differentiate
7:45
Example II: Differentiate
12:38
Example III: Differentiate
15:13
Example IV: Differentiate
16:20
Example V: Differentiate
19:19
Example VI: Find the Equation of the Tangent Line to a Function at a Given Point
12:18
Example VII: Find the First & Second Derivatives
25:59
Example VIII
27:47
Part A: Find the Velocity & Acceleration Functions as Functions of t
27:48
Part B: Find the Acceleration after 3 Seconds
30:12
Part C: Find the Acceleration when the Velocity is 0
30:53
Part D: Graph the Position, Velocity, & Acceleration Graphs
32:50
Example IX: Find a Cubic Function Whose Graph has Horizontal Tangents
34:53
Example X: Find a Point on a Graph
42:31
The Product, Power & Quotient Rules

47m 25s

Intro
0:00
The Product, Power and Quotient Rules
0:19
Differentiate Functions
0:20
Product Rule
5:30
Quotient Rule
9:15
Power Rule
10:00
Example I: Product Rule
13:48
Example II: Quotient Rule
16:13
Example III: Power Rule
18:28
Example IV: Find dy/dx
19:57
Example V: Find dy/dx
24:53
Example VI: Find dy/dx
28:38
Example VII: Find an Equation for the Tangent to the Curve
34:54
Example VIII: Find d²y/dx²
38:08
Derivatives of the Trigonometric Functions

41m 8s

Intro
0:00
Derivatives of the Trigonometric Functions
0:09
Let's Find the Derivative of f(x) = sin x
0:10
Important Limits to Know
4:59
d/dx (sin x)
6:06
d/dx (cos x)
6:38
d/dx (tan x)
6:50
d/dx (csc x)
7:02
d/dx (sec x)
7:15
d/dx (cot x)
7:27
Example I: Differentiate f(x) = x² - 4 cos x
7:56
Example II: Differentiate f(x) = x⁵ tan x
9:04
Example III: Differentiate f(x) = (cos x) / (3 + sin x)
10:56
Example IV: Differentiate f(x) = e^x / (tan x - sec x)
14:06
Example V: Differentiate f(x) = (csc x - 4) / (cot x)
15:37
Example VI: Find an Equation of the Tangent Line
21:48
Example VII: For What Values of x Does the Graph of the Function x + 3 cos x Have a Horizontal Tangent?
25:17
28:23
Example IX: Evaluate
33:22
Example X: Evaluate
36:38
The Chain Rule

24m 56s

Intro
0:00
The Chain Rule
0:13
Recall the Composite Functions
0:14
Derivatives of Composite Functions
1:34
Example I: Identify f(x) and g(x) and Differentiate
6:41
Example II: Identify f(x) and g(x) and Differentiate
9:47
Example III: Differentiate
11:03
Example IV: Differentiate f(x) = -5 / (x² + 3)³
12:15
Example V: Differentiate f(x) = cos(x² + c²)
14:35
Example VI: Differentiate f(x) = cos⁴x +c²
15:41
Example VII: Differentiate
17:03
Example VIII: Differentiate f(x) = sin(tan x²)
19:01
Example IX: Differentiate f(x) = sin(tan² x)
21:02
More Chain Rule Example Problems

25m 32s

Intro
0:00
Example I: Differentiate f(x) = sin(cos(tanx))
0:38
Example II: Find an Equation for the Line Tangent to the Given Curve at the Given Point
2:25
Example III: F(x) = f(g(x)), Find F' (6)
4:22
Example IV: Differentiate & Graph both the Function & the Derivative in the Same Window
5:35
Example V: Differentiate f(x) = ( (x-8)/(x+3) )⁴
10:18
Example VI: Differentiate f(x) = sec²(12x)
12:28
Example VII: Differentiate
14:41
Example VIII: Differentiate
19:25
Example IX: Find an Expression for the Rate of Change of the Volume of the Balloon with Respect to Time
21:13
Implicit Differentiation

52m 31s

Intro
0:00
Implicit Differentiation
0:09
Implicit Differentiation
0:10
Example I: Find (dy)/(dx) by both Implicit Differentiation and Solving Explicitly for y
12:15
Example II: Find (dy)/(dx) of x³ + x²y + 7y² = 14
19:18
Example III: Find (dy)/(dx) of x³y² + y³x² = 4x
21:43
Example IV: Find (dy)/(dx) of the Following Equation
24:13
Example V: Find (dy)/(dx) of 6sin x cos y = 1
29:00
Example VI: Find (dy)/(dx) of x² cos² y + y sin x = 2sin x cos y
31:02
Example VII: Find (dy)/(dx) of √(xy) = 7 + y²e^x
37:36
Example VIII: Find (dy)/(dx) of 4(x²+y²)² = 35(x²-y²)
41:03
Example IX: Find (d²y)/(dx²) of x² + y² = 25
44:05
Example X: Find (d²y)/(dx²) of sin x + cos y = sin(2x)
47:48
Section 3: Applications of the Derivative
Linear Approximations & Differentials

47m 34s

Intro
0:00
Linear Approximations & Differentials
0:09
Linear Approximations & Differentials
0:10
Example I: Linear Approximations & Differentials
11:27
Example II: Linear Approximations & Differentials
20:19
Differentials
30:32
Differentials
30:33
Example III: Linear Approximations & Differentials
34:09
Example IV: Linear Approximations & Differentials
35:57
Example V: Relative Error
38:46
Related Rates

45m 33s

Intro
0:00
Related Rates
0:08
Strategy for Solving Related Rates Problems #1
0:09
Strategy for Solving Related Rates Problems #2
1:46
Strategy for Solving Related Rates Problems #3
2:06
Strategy for Solving Related Rates Problems #4
2:50
Strategy for Solving Related Rates Problems #5
3:38
Example I: Radius of a Balloon
5:15
12:52
Example III: Water Tank
19:08
Example IV: Distance between Two Cars
29:27
Example V: Line-of-Sight
36:20
More Related Rates Examples

37m 17s

Intro
0:00
0:14
Example II: Particle
4:45
Example III: Water Level
10:28
Example IV: Clock
20:47
Example V: Distance between a House and a Plane
29:11
Maximum & Minimum Values of a Function

40m 44s

Intro
0:00
Maximum & Minimum Values of a Function, Part 1
0:23
Absolute Maximum
2:20
Absolute Minimum
2:52
Local Maximum
3:38
Local Minimum
4:26
Maximum & Minimum Values of a Function, Part 2
6:11
Function with Absolute Minimum but No Absolute Max, Local Max, and Local Min
7:18
Function with Local Max & Min but No Absolute Max & Min
8:48
Formal Definitions
10:43
Absolute Maximum
11:18
Absolute Minimum
12:57
Local Maximum
14:37
Local Minimum
16:25
Extreme Value Theorem
18:08
Theorem: f'(c) = 0
24:40
Critical Number (Critical Value)
26:14
Procedure for Finding the Critical Values of f(x)
28:32
Example I: Find the Critical Values of f(x) x + sinx
29:51
Example II: What are the Absolute Max & Absolute Minimum of f(x) = x + 4 sinx on [0,2π]
35:31
Example Problems for Max & Min

40m 44s

Intro
0:00
Example I: Identify Absolute and Local Max & Min on the Following Graph
0:11
Example II: Sketch the Graph of a Continuous Function
3:11
Example III: Sketch the Following Graphs
4:40
Example IV: Find the Critical Values of f (x) = 3x⁴ - 7x³ + 4x²
6:13
Example V: Find the Critical Values of f(x) = |2x - 5|
8:42
Example VI: Find the Critical Values
11:42
Example VII: Find the Critical Values f(x) = cos²(2x) on [0,2π]
16:57
Example VIII: Find the Absolute Max & Min f(x) = 2sinx + 2cos x on [0,(π/3)]
20:08
Example IX: Find the Absolute Max & Min f(x) = (ln(2x)) / x on [1,3]
24:39
The Mean Value Theorem

25m 54s

Intro
0:00
Rolle's Theorem
0:08
Rolle's Theorem: If & Then
0:09
Rolle's Theorem: Geometrically
2:06
There May Be More than 1 c Such That f'( c ) = 0
3:30
Example I: Rolle's Theorem
4:58
The Mean Value Theorem
9:12
The Mean Value Theorem: If & Then
9:13
The Mean Value Theorem: Geometrically
11:07
Example II: Mean Value Theorem
13:43
Example III: Mean Value Theorem
21:19
Using Derivatives to Graph Functions, Part I

25m 54s

Intro
0:00
Using Derivatives to Graph Functions, Part I
0:12
Increasing/ Decreasing Test
0:13
Example I: Find the Intervals Over Which the Function is Increasing & Decreasing
3:26
Example II: Find the Local Maxima & Minima of the Function
19:18
Example III: Find the Local Maxima & Minima of the Function
31:39
Using Derivatives to Graph Functions, Part II

44m 58s

Intro
0:00
Using Derivatives to Graph Functions, Part II
0:13
Concave Up & Concave Down
0:14
What Does This Mean in Terms of the Derivative?
6:14
Point of Inflection
8:52
Example I: Graph the Function
13:18
Example II: Function x⁴ - 5x²
19:03
Intervals of Increase & Decrease
19:04
Local Maxes and Mins
25:01
Intervals of Concavity & X-Values for the Points of Inflection
29:18
Intervals of Concavity & Y-Values for the Points of Inflection
34:18
Graphing the Function
40:52
Example Problems I

49m 19s

Intro
0:00
Example I: Intervals, Local Maxes & Mins
0:26
Example II: Intervals, Local Maxes & Mins
5:05
Example III: Intervals, Local Maxes & Mins, and Inflection Points
13:40
Example IV: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity
23:02
Example V: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity
34:36
Example Problems III

59m 1s

Intro
0:00
Example I: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
0:11
Example II: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
21:24
Example III: Cubic Equation f(x) = Ax³ + Bx² + Cx + D
37:56
Example IV: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
46:19
L'Hospital's Rule

30m 9s

Intro
0:00
L'Hospital's Rule
0:19
Indeterminate Forms
0:20
L'Hospital's Rule
3:38
Example I: Evaluate the Following Limit Using L'Hospital's Rule
8:50
Example II: Evaluate the Following Limit Using L'Hospital's Rule
10:30
Indeterminate Products
11:54
Indeterminate Products
11:55
Example III: L'Hospital's Rule & Indeterminate Products
13:57
Indeterminate Differences
17:00
Indeterminate Differences
17:01
Example IV: L'Hospital's Rule & Indeterminate Differences
18:57
Indeterminate Powers
22:20
Indeterminate Powers
22:21
Example V: L'Hospital's Rule & Indeterminate Powers
25:13
Example Problems for L'Hospital's Rule

38m 14s

Intro
0:00
Example I: Evaluate the Following Limit
0:17
Example II: Evaluate the Following Limit
2:45
Example III: Evaluate the Following Limit
6:54
Example IV: Evaluate the Following Limit
8:43
Example V: Evaluate the Following Limit
11:01
Example VI: Evaluate the Following Limit
14:48
Example VII: Evaluate the Following Limit
17:49
Example VIII: Evaluate the Following Limit
20:37
Example IX: Evaluate the Following Limit
25:16
Example X: Evaluate the Following Limit
32:44
Optimization Problems I

49m 59s

Intro
0:00
Example I: Find the Dimensions of the Box that Gives the Greatest Volume
1:23
Fundamentals of Optimization Problems
18:08
Fundamental #1
18:33
Fundamental #2
19:09
Fundamental #3
19:19
Fundamental #4
20:59
Fundamental #5
21:55
Fundamental #6
23:44
Example II: Demonstrate that of All Rectangles with a Given Perimeter, the One with the Largest Area is a Square
24:36
Example III: Find the Points on the Ellipse 9x² + y² = 9 Farthest Away from the Point (1,0)
35:13
Example IV: Find the Dimensions of the Rectangle of Largest Area that can be Inscribed in a Circle of Given Radius R
43:10
Optimization Problems II

55m 10s

Intro
0:00
Example I: Optimization Problem
0:13
Example II: Optimization Problem
17:34
Example III: Optimization Problem
35:06
Example IV: Revenue, Cost, and Profit
43:22
Newton's Method

30m 22s

Intro
0:00
Newton's Method
0:45
Newton's Method
0:46
Example I: Find x2 and x3
13:18
Example II: Use Newton's Method to Approximate
15:48
Example III: Find the Root of the Following Equation to 6 Decimal Places
19:57
Example IV: Use Newton's Method to Find the Coordinates of the Inflection Point
23:11
Section 4: Integrals
Antiderivatives

55m 26s

Intro
0:00
Antiderivatives
0:23
Definition of an Antiderivative
0:24
Antiderivative Theorem
7:58
Function & Antiderivative
12:10
x^n
12:30
1/x
13:00
e^x
13:08
cos x
13:18
sin x
14:01
sec² x
14:11
secxtanx
14:18
1/√(1-x²)
14:26
1/(1+x²)
14:36
-1/√(1-x²)
14:45
Example I: Find the Most General Antiderivative for the Following Functions
15:07
Function 1: f(x) = x³ -6x² + 11x - 9
15:42
Function 2: f(x) = 14√(x) - 27 4√x
19:12
Function 3: (fx) = cos x - 14 sinx
20:53
Function 4: f(x) = (x⁵+2√x )/( x^(4/3) )
22:10
Function 5: f(x) = (3e^x) - 2/(1+x²)
25:42
Example II: Given the Following, Find the Original Function f(x)
26:37
Function 1: f'(x) = 5x³ - 14x + 24, f(2) = 40
27:55
Function 2: f'(x) 3 sinx + sec²x, f(π/6) = 5
30:34
Function 3: f''(x) = 8x - cos x, f(1.5) = 12.7, f'(1.5) = 4.2
32:54
Function 4: f''(x) = 5/(√x), f(2) 15, f'(2) = 7
37:54
Example III: Falling Object
41:58
Problem 1: Find an Equation for the Height of the Ball after t Seconds
42:48
Problem 2: How Long Will It Take for the Ball to Strike the Ground?
48:30
Problem 3: What is the Velocity of the Ball as it Hits the Ground?
49:52
Problem 4: Initial Velocity of 6 m/s, How Long Does It Take to Reach the Ground?
50:46
The Area Under a Curve

51m 3s

Intro
0:00
The Area Under a Curve
0:13
Approximate Using Rectangles
0:14
Let's Do This Again, Using 4 Different Rectangles
9:40
Approximate with Rectangles
16:10
Left Endpoint
18:08
Right Endpoint
25:34
Left Endpoint vs. Right Endpoint
30:58
Number of Rectangles
34:08
True Area
37:36
True Area
37:37
Sigma Notation & Limits
43:32
When You Have to Explicitly Solve Something
47:56
Example Problems for Area Under a Curve

33m 7s

Intro
0:00
Example I: Using Left Endpoint & Right Endpoint to Approximate Area Under a Curve
0:10
Example II: Using 5 Rectangles, Approximate the Area Under the Curve
11:32
Example III: Find the True Area by Evaluating the Limit Expression
16:07
Example IV: Find the True Area by Evaluating the Limit Expression
24:52
The Definite Integral

43m 19s

Intro
0:00
The Definite Integral
0:08
Definition to Find the Area of a Curve
0:09
Definition of the Definite Integral
4:08
Symbol for Definite Integral
8:45
Regions Below the x-axis
15:18
Associating Definite Integral to a Function
19:38
Integrable Function
27:20
Evaluating the Definite Integral
29:26
Evaluating the Definite Integral
29:27
Properties of the Definite Integral
35:24
Properties of the Definite Integral
35:25
Example Problems for The Definite Integral

32m 14s

Intro
0:00
Example I: Approximate the Following Definite Integral Using Midpoints & Sub-intervals
0:11
Example II: Express the Following Limit as a Definite Integral
5:28
Example III: Evaluate the Following Definite Integral Using the Definition
6:28
Example IV: Evaluate the Following Integral Using the Definition
17:06
Example V: Evaluate the Following Definite Integral by Using Areas
25:41
Example VI: Definite Integral
30:36
The Fundamental Theorem of Calculus

24m 17s

Intro
0:00
The Fundamental Theorem of Calculus
0:17
Evaluating an Integral
0:18
Lim as x → ∞
12:19
Taking the Derivative
14:06
Differentiation & Integration are Inverse Processes
15:04
1st Fundamental Theorem of Calculus
20:08
1st Fundamental Theorem of Calculus
20:09
2nd Fundamental Theorem of Calculus
22:30
2nd Fundamental Theorem of Calculus
22:31
Example Problems for the Fundamental Theorem

25m 21s

Intro
0:00
Example I: Find the Derivative of the Following Function
0:17
Example II: Find the Derivative of the Following Function
1:40
Example III: Find the Derivative of the Following Function
2:32
Example IV: Find the Derivative of the Following Function
5:55
Example V: Evaluate the Following Integral
7:13
Example VI: Evaluate the Following Integral
9:46
Example VII: Evaluate the Following Integral
12:49
Example VIII: Evaluate the Following Integral
13:53
Example IX: Evaluate the Following Graph
15:24
Local Maxs and Mins for g(x)
15:25
Where Does g(x) Achieve Its Absolute Max on [0,8]
20:54
On What Intervals is g(x) Concave Up/Down?
22:20
Sketch a Graph of g(x)
24:34
More Example Problems, Including Net Change Applications

34m 22s

Intro
0:00
Example I: Evaluate the Following Indefinite Integral
0:10
Example II: Evaluate the Following Definite Integral
0:59
Example III: Evaluate the Following Integral
2:59
Example IV: Velocity Function
7:46
Part A: Net Displacement
7:47
Part B: Total Distance Travelled
13:15
Example V: Linear Density Function
20:56
Example VI: Acceleration Function
25:10
Part A: Velocity Function at Time t
25:11
Part B: Total Distance Travelled During the Time Interval
28:38
Solving Integrals by Substitution

27m 20s

Intro
0:00
Table of Integrals
0:35
Example I: Evaluate the Following Indefinite Integral
2:02
Example II: Evaluate the Following Indefinite Integral
7:27
Example IIII: Evaluate the Following Indefinite Integral
10:57
Example IV: Evaluate the Following Indefinite Integral
12:33
Example V: Evaluate the Following
14:28
Example VI: Evaluate the Following
16:00
Example VII: Evaluate the Following
19:01
Example VIII: Evaluate the Following
21:49
Example IX: Evaluate the Following
24:34
Section 5: Applications of Integration
Areas Between Curves

34m 56s

Intro
0:00
Areas Between Two Curves: Function of x
0:08
Graph 1: Area Between f(x) & g(x)
0:09
Graph 2: Area Between f(x) & g(x)
4:07
Is It Possible to Write as a Single Integral?
8:20
Area Between the Curves on [a,b]
9:24
Absolute Value
10:32
Formula for Areas Between Two Curves: Top Function - Bottom Function
17:03
Areas Between Curves: Function of y
17:49
What if We are Given Functions of y?
17:50
Formula for Areas Between Two Curves: Right Function - Left Function
21:48
Finding a & b
22:32
Example Problems for Areas Between Curves

42m 55s

Intro
0:00
Instructions for the Example Problems
0:10
Example I: y = 7x - x² and y=x
0:37
Example II: x=y²-3, x=e^((1/2)y), y=-1, and y=2
6:25
Example III: y=(1/x), y=(1/x³), and x=4
12:25
Example IV: 15-2x² and y=x²-5
15:52
Example V: x=(1/8)y³ and x=6-y²
20:20
Example VI: y=cos x, y=sin(2x), [0,π/2]
24:34
Example VII: y=2x², y=10x², 7x+2y=10
29:51
Example VIII: Velocity vs. Time
33:23
Part A: At 2.187 Minutes, Which care is Further Ahead?
33:24
Part B: If We Shaded the Region between the Graphs from t=0 to t=2.187, What Would This Shaded Area Represent?
36:32
Part C: At 4 Minutes Which Car is Ahead?
37:11
Part D: At What Time Will the Cars be Side by Side?
37:50
Volumes I: Slices

34m 15s

Intro
0:00
Volumes I: Slices
0:18
Rotate the Graph of y=√x about the x-axis
0:19
How can I use Integration to Find the Volume?
3:16
Slice the Solid Like a Loaf of Bread
5:06
Volumes Definition
8:56
Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
12:18
Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
19:05
Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
25:28
Volumes II: Volumes by Washers

51m 43s

Intro
0:00
Volumes II: Volumes by Washers
0:11
Rotating Region Bounded by y=x³ & y=x around the x-axis
0:12
Equation for Volumes by Washer
11:14
Process for Solving Volumes by Washer
13:40
Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
15:58
Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
25:07
Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
34:20
Example IV: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
44:05
Volumes III: Solids That Are Not Solids-of-Revolution

49m 36s

Intro
0:00
Solids That Are Not Solids-of-Revolution
0:11
Cross-Section Area Review
0:12
Cross-Sections That Are Not Solids-of-Revolution
7:36
Example I: Find the Volume of a Pyramid Whose Base is a Square of Side-length S, and Whose Height is H
10:54
Example II: Find the Volume of a Solid Whose Cross-sectional Areas Perpendicular to the Base are Equilateral Triangles
20:39
Example III: Find the Volume of a Pyramid Whose Base is an Equilateral Triangle of Side-Length A, and Whose Height is H
29:27
Example IV: Find the Volume of a Solid Whose Base is Given by the Equation 16x² + 4y² = 64
36:47
Example V: Find the Volume of a Solid Whose Base is the Region Bounded by the Functions y=3-x² and the x-axis
46:13
Volumes IV: Volumes By Cylindrical Shells

50m 2s

Intro
0:00
Volumes by Cylindrical Shells
0:11
Find the Volume of the Following Region
0:12
Volumes by Cylindrical Shells: Integrating Along x
14:12
Volumes by Cylindrical Shells: Integrating Along y
14:40
Volumes by Cylindrical Shells Formulas
16:22
Example I: Using the Method of Cylindrical Shells, Find the Volume of the Solid
18:33
Example II: Using the Method of Cylindrical Shells, Find the Volume of the Solid
25:57
Example III: Using the Method of Cylindrical Shells, Find the Volume of the Solid
31:38
Example IV: Using the Method of Cylindrical Shells, Find the Volume of the Solid
38:44
Example V: Using the Method of Cylindrical Shells, Find the Volume of the Solid
44:03
The Average Value of a Function

32m 13s

Intro
0:00
The Average Value of a Function
0:07
Average Value of f(x)
0:08
What if The Domain of f(x) is Not Finite?
2:23
Let's Calculate Average Value for f(x) = x² [2,5]
4:46
Mean Value Theorem for Integrate
9:25
Example I: Find the Average Value of the Given Function Over the Given Interval
14:06
Example II: Find the Average Value of the Given Function Over the Given Interval
18:25
Example III: Find the Number A Such that the Average Value of the Function f(x) = -4x² + 8x + 4 Equals 2 Over the Interval [-1,A]
24:04
Example IV: Find the Average Density of a Rod
27:47
Section 6: Techniques of Integration
Integration by Parts

50m 32s

Intro
0:00
Integration by Parts
0:08
The Product Rule for Differentiation
0:09
Integrating Both Sides Retains the Equality
0:52
Differential Notation
2:24
Example I: ∫ x cos x dx
5:41
Example II: ∫ x² sin(2x)dx
12:01
Example III: ∫ (e^x) cos x dx
18:19
Example IV: ∫ (sin^-1) (x) dx
23:42
Example V: ∫₁⁵ (lnx)² dx
28:25
Summary
32:31
Tabular Integration
35:08
Case 1
35:52
Example: ∫x³sinx dx
36:39
Case 2
40:28
Example: ∫e^(2x) sin 3x
41:14
Trigonometric Integrals I

24m 50s

Intro
0:00
Example I: ∫ sin³ (x) dx
1:36
Example II: ∫ cos⁵(x)sin²(x)dx
4:36
Example III: ∫ sin⁴(x)dx
9:23
Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sin^m) (x) (cos^p) (x) dx
15:59
#1: Power of sin is Odd
16:00
#2: Power of cos is Odd
16:41
#3: Powers of Both sin and cos are Odd
16:55
#4: Powers of Both sin and cos are Even
17:10
Example IV: ∫ tan⁴ (x) sec⁴ (x) dx
17:34
Example V: ∫ sec⁹(x) tan³(x) dx
20:55
Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sec^m) (x) (tan^p) (x) dx
23:31
#1: Power of sec is Odd
23:32
#2: Power of tan is Odd
24:04
#3: Powers of sec is Odd and/or Power of tan is Even
24:18
Trigonometric Integrals II

22m 12s

Intro
0:00
Trigonometric Integrals II
0:09
Recall: ∫tanx dx
0:10
Let's Find ∫secx dx
3:23
Example I: ∫ tan⁵ (x) dx
6:23
Example II: ∫ sec⁵ (x) dx
11:41
Summary: How to Deal with Integrals of Different Types
19:04
Identities to Deal with Integrals of Different Types
19:05
Example III: ∫cos(5x)sin(9x)dx
19:57
More Example Problems for Trigonometric Integrals

17m 22s

Intro
0:00
Example I: ∫sin²(x)cos⁷(x)dx
0:14
Example II: ∫x sin²(x) dx
3:56
Example III: ∫csc⁴ (x/5)dx
8:39
Example IV: ∫( (1-tan²x)/(sec²x) ) dx
11:17
Example V: ∫ 1 / (sinx-1) dx
13:19
Integration by Partial Fractions I

55m 12s

Intro
0:00
Integration by Partial Fractions I
0:11
Recall the Idea of Finding a Common Denominator
0:12
Decomposing a Rational Function to Its Partial Fractions
4:10
2 Types of Rational Function: Improper & Proper
5:16
Improper Rational Function
7:26
Improper Rational Function
7:27
Proper Rational Function
11:16
Proper Rational Function & Partial Fractions
11:17
Linear Factors
14:04
15:02
Case 1: G(x) is a Product of Distinct Linear Factors
17:10
Example I: Integration by Partial Fractions
20:33
Case 2: D(x) is a Product of Linear Factors
40:58
Example II: Integration by Partial Fractions
44:41
Integration by Partial Fractions II

42m 57s

Intro
0:00
Case 3: D(x) Contains Irreducible Factors
0:09
Example I: Integration by Partial Fractions
5:19
Example II: Integration by Partial Fractions
16:22
Case 4: D(x) has Repeated Irreducible Quadratic Factors
27:30
Example III: Integration by Partial Fractions
30:19
Section 7: Differential Equations
Introduction to Differential Equations

46m 37s

Intro
0:00
Introduction to Differential Equations
0:09
Overview
0:10
Differential Equations Involving Derivatives of y(x)
2:08
Differential Equations Involving Derivatives of y(x) and Function of y(x)
3:23
Equations for an Unknown Number
6:28
What are These Differential Equations Saying?
10:30
Verifying that a Function is a Solution of the Differential Equation
13:00
Verifying that a Function is a Solution of the Differential Equation
13:01
Verify that y(x) = 4e^x + 3x² + 6x + e^π is a Solution of this Differential Equation
17:20
General Solution
22:00
Particular Solution
24:36
Initial Value Problem
27:42
Example I: Verify that a Family of Functions is a Solution of the Differential Equation
32:24
Example II: For What Values of K Does the Function Satisfy the Differential Equation
36:07
Example III: Verify the Solution and Solve the Initial Value Problem
39:47
Separation of Variables

28m 8s

Intro
0:00
Separation of Variables
0:28
Separation of Variables
0:29
Example I: Solve the Following g Initial Value Problem
8:29
Example II: Solve the Following g Initial Value Problem
13:46
Example III: Find an Equation of the Curve
18:48
Population Growth: The Standard & Logistic Equations

51m 7s

Intro
0:00
Standard Growth Model
0:30
Definition of the Standard/Natural Growth Model
0:31
Initial Conditions
8:00
The General Solution
9:16
Example I: Standard Growth Model
10:45
Logistic Growth Model
18:33
Logistic Growth Model
18:34
Solving the Initial Value Problem
25:21
What Happens When t → ∞
36:42
Example II: Solve the Following g Initial Value Problem
41:50
Relative Growth Rate
46:56
Relative Growth Rate
46:57
Relative Growth Rate Version for the Standard model
49:04
Slope Fields

24m 37s

Intro
0:00
Slope Fields
0:35
Slope Fields
0:36
Graphing the Slope Fields, Part 1
11:12
Graphing the Slope Fields, Part 2
15:37
Graphing the Slope Fields, Part 3
17:25
Steps to Solving Slope Field Problems
20:24
Example I: Draw or Generate the Slope Field of the Differential Equation y'=x cos y
22:38
Section 8: AP Practic Exam
AP Practice Exam: Section 1, Part A No Calculator

45m 29s

Intro
0:00
0:10
Problem #1
1:26
Problem #2
2:52
Problem #3
4:42
Problem #4
7:03
Problem #5
10:01
Problem #6
13:49
Problem #7
15:16
Problem #8
19:06
Problem #9
23:10
Problem #10
28:10
Problem #11
31:30
Problem #12
33:53
Problem #13
37:45
Problem #14
41:17
AP Practice Exam: Section 1, Part A No Calculator, cont.

41m 55s

Intro
0:00
Problem #15
0:22
Problem #16
3:10
Problem #17
5:30
Problem #18
8:03
Problem #19
9:53
Problem #20
14:51
Problem #21
17:30
Problem #22
22:12
Problem #23
25:48
Problem #24
29:57
Problem #25
33:35
Problem #26
35:57
Problem #27
37:57
Problem #28
40:04
AP Practice Exam: Section I, Part B Calculator Allowed

58m 47s

Intro
0:00
Problem #1
1:22
Problem #2
4:55
Problem #3
10:49
Problem #4
13:05
Problem #5
14:54
Problem #6
17:25
Problem #7
18:39
Problem #8
20:27
Problem #9
26:48
Problem #10
28:23
Problem #11
34:03
Problem #12
36:25
Problem #13
39:52
Problem #14
43:12
Problem #15
47:18
Problem #16
50:41
Problem #17
56:38
AP Practice Exam: Section II, Part A Calculator Allowed

25m 40s

Intro
0:00
Problem #1: Part A
1:14
Problem #1: Part B
4:46
Problem #1: Part C
8:00
Problem #2: Part A
12:24
Problem #2: Part B
16:51
Problem #2: Part C
17:17
Problem #3: Part A
18:16
Problem #3: Part B
19:54
Problem #3: Part C
21:44
Problem #3: Part D
22:57
AP Practice Exam: Section II, Part B No Calculator

31m 20s

Intro
0:00
Problem #4: Part A
1:35
Problem #4: Part B
5:54
Problem #4: Part C
8:50
Problem #4: Part D
9:40
Problem #5: Part A
11:26
Problem #5: Part B
13:11
Problem #5: Part C
15:07
Problem #5: Part D
19:57
Problem #6: Part A
22:01
Problem #6: Part B
25:34
Problem #6: Part C
28:54

• ## Transcription

 3 answersLast reply by: Professor HovasapianMon Jan 29, 2018 11:15 PMPost by Magic Fu on January 23, 2018can you go over Euler's method? 1 answerLast reply by: Professor HovasapianWed Oct 26, 2016 7:42 PMPost by Tiffany Warner on September 28, 2016Hello Professor Hovasapian,With Example 1, I am really confused. In the last line of solving, a C reappears in the problem. x((-3/x^4)+(5/4)) becomes (-3C/x^3) + (5x/4). How come the C got thrown in? I'm probably missing something obvious but I keep looking it over and I'm not seeing it. Thank you. 3 answersLast reply by: Professor HovasapianMon Jul 25, 2016 7:07 PMPost by Peter Ke on July 23, 2016At 40:40, how is this ----> http://prnt.sc/bwjrof the derivative of y?I thought it was:http://prnt.sc/bwjrofPlease explain.

### Introduction to Differential Equations

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Introduction to Differential Equations 0:09
• Overview
• Differential Equations Involving Derivatives of y(x)
• Differential Equations Involving Derivatives of y(x) and Function of y(x)
• Equations for an Unknown Number
• What are These Differential Equations Saying?
• Verifying that a Function is a Solution of the Differential Equation 13:00
• Verifying that a Function is a Solution of the Differential Equation
• Verify that y(x) = 4e^x + 3x² + 6x + e^π is a Solution of this Differential Equation
• General Solution
• Particular Solution
• Initial Value Problem
• Example I: Verify that a Family of Functions is a Solution of the Differential Equation 32:24
• Example II: For What Values of K Does the Function Satisfy the Differential Equation 36:07
• Example III: Verify the Solution and Solve the Initial Value Problem 39:47

### Transcription: Introduction to Differential Equations

Hello, welcome back to www.educator.com, and welcome back to AP Calculus.0000

Today, we are going to start our discussion of differential equations.0004

Let us jump right on in.0008

Let us going to start with what a differential equation is, it is very simple.0013

A big deal was made about differential equations, I’m not exactly sure why.0017

It is just one of those things that has become sort of mythical, like they are difficult crazy, insane, esoteric things.0021

They are not, it is just an equation like any other equation.0028

Except instead of solving an algebraic equation where you are searching for a number x = 5,0031

here the unknown is actually just a function.0037

We are trying to recover a function that we do not know what it is, that is all.0040

I will stick with black here.0046

A differential equation which we will often just call a de, is an equation that involves one or more derivatives of an unknown function y(x).0048

That is it, that is all a different equation is.0101

It is just an equation that involves one or more derivatives of an unknown function x and it can look like anything.0103

It can look like anything.0113

Sometimes it involves only the derivatives of y(x).0123

Let me go on to blue, actually.0128

Sometimes it involves only the derivatives, and of course, other functions of x, other constant thing like that.0131

Only the derivatives of y(x).0147

The examples would be something like dy dx - 4x = 3x, that is a differential equation.0155

It involves the derivative of an unknown function.0167

y is the functional that we are trying to recover, y(x) = sin(3x)² + 9, something like that, whatever it is.0170

That is what we are trying to elucidate.0178

We might have d² y dx², it might involve a second derivative, + 6x = dy dx.0181

Now the second derivative and the first derivative are involved in the equation.0192

How do we solve this to recover the function y(x).0195

These are two differential equations that involve only the derivatives of y(x).0199

Sometimes the de involves derivatives of y(x), as well as the function y itself which we do not know.0207

We just call it y.0231

As well as the function y(x) itself.0234

Some examples of that are dy dx = x × y(x).0244

Or we might have dy dx² – dy dx = y(x) × sin of y(x).0256

Here we have the derivative, as well as the unknown function itself.0276

Here we have the first derivative² - the first derivative.0283

It is telling you that when I do this, when I take the first derivative² of whatever function it is and0288

if I subtract the first derivative from that, it actually equal the original function × the sin of the original function.0295

That is it, that is all this says.0303

It is just an equation that involves the derivatives of an unknown function.0304

Sometimes the function itself, sometimes not.0308

It is an equation, we are solving for y.0311

We are trying to find the function y(x) = something, that is all we are doing.0314

Once again, our job is to find the unknown function y(x).0325

We are just trying to find a function, y = x², y = sin x, y = e ⁺x, whatever it is.0343

Let us see what we can do.0358

Do I want to write this thing?0362

I might as well, I have it here.0367

Again, to reiterate, for years, we have been solving algebraic and trigonometric functions for an unknown number.0369

Every time, all these years in math, we have been trying to solve for x or t, or z, whatever it is, for unknown variable, for an unknown number.0398

In other words, the variable x is going to be some unknown number.0412

We have something like x² + 4x + 5 = 0.0415

x is equal to this, this, this, whatever it is.0421

Or we have 2 sin x + cos x is equal to 3.0424

Now our unknown is not a number, it is an actual function.0431

Now our unknown is a function y(x).0441

When writing a differential equation, we often use the prime notation instead of a dy dx and dy dt notation.0456

We often you the prime notation of differentiation.0472

There is a prime notation and we would leave off the independent variable, usually x.0481

You leave off the independent variable from the notation y(x).0500

We do not really write y(x), we will just write y for the function.0509

Our sample equations, the four equations that we mentioned on the first page, would look like this.0520

Our sample equations would look like y’ – 4x² = 3x, y” + 6x = y’, y’ = xy, y’² – y’ = y sin y.0532

It is a lot cleaner to use this prime notation.0571

Now take note, y’² is not the same as y”.0575

y” is the second derivative.0591

y’² is the first derivative², be very careful with that.0594

Always remember, I will repeat this several times.0601

y is a function not a number.0606

We are looking for a function of x.0617

The question is, what are these differential equation is saying, what do they mean?0626

What are these de is saying?0634

If I have something, let us take the first equation y’ - 4x² = 3x.0647

What does this equation mean?0657

This means that y(x) is a function such that, when I take the first derivative then subtract 4x², I am left with 3x.0662

That is all this is saying.0718

A differential equation establishes a relationship between the derivative and other things.0720

Those other things might be functions of x.0726

They might be the original function y(x) itself, that is all it is.0728

That is all any equation is.0732

It is a relationship between the different parts.0733

In this particular case, let us say some scientist collected some data.0736

When they analyze that data, they found a relationship between the rate of change of y.0741

That is what a derivative is, it is a rate of change.0746

The rate of change of y - 4x² actually ends up being equal to 3x.0750

We want to find out what y is, what function satisfies this differential equation?0756

What function, when I take the first derivative, subtract 4x² from it will actually give me 3x?0762

That is what we are doing, that all we are doing.0768

We are trying to find a function that actually does this.0771

Let us go to, I’m going to try a function, I’m going to ask you to verify that it actually solve the differential equation.0777

Verify that y(x) = 4/3 x³ + 3/2 x² + 19, is a solution of the de that we are just working with, which is y’ – 4x² = 3x.0790

Verify that this function that I gave you is a solution of the de.0825

How do you verify that a particular function is a solution of a different an equation?0832

You take the derivatives, you plug it into the equation, and you will see if the left side actually equals the right side.0837

This equality, verify it just like we did in trig identities.0845

You are actually verifying that the left side = the right side.0848

That is how you do it.0853

You take derivatives, however many you need, whatever the differential equations says.0855

Take derivatives and substitute into the de to check that equality holds.0871

In this particular case, this is the function, so I find y’.0896

y’ is going to equal 4x² + 3x.0901

Now I substitute this into the differential equation.0914

We have y’ which is equal to 4x² + 3x - 4x².0925

The questions is, does is equal 3x?0936

4x² - 4x², you get 3x = 3x.0939

Yes, this confirms that this is a solution of that differential equation.0944

Let us take a look at the second differential equation, that was y’’ + 6x = y’.0956

What is this one saying?0969

This says y(x) is such a function that differentiating twice then adding a 6x,0977

actually gives you the first derivative of the function.1011

This is saying find the function y(x), that when you take the second derivative of it and then you add 6x to it,1022

you actually end up getting the first derivative of the function.1029

What function satisfies that equality?1033

Here is the verification.1037

I put it to you, verify that the function y(x) which is equal to 4e ⁺x + 3x² + 6x + e ⁺π is a solution of this differential equation.1041

Verify that.1073

How do you verify it?1074

You take derivatives, you plug it back in, and if you see if the left side = the right side.1075

Let us go ahead and do that.1081

This is our y, let us go ahead and take the first derivative.1082

y’ is equal to 4e ⁺x + 6x + 6 y”.1086

Because it involves double prime, it is equal to 4e ⁺x + 6.1096

Now I substitute.1105

Now put these into the differential equation to see if the equality holds.1116

y” that is going to be 4e ⁺x + 6, that is the y” part, and then, + 6x.1129

The question is, does it equal the first derivative which is 4e ⁺x + 6x + 6.1145

Yes, this is a solution of that differential equation.1157

I have verified it by taking derivatives, plugging it in and showing that the left side = the right side.1168

Let us do one more of these.1180

Let us try the third differential equation.1183

The third differential equation was y’ = xy.1188

This says, when I differentiate y one time, when I differentiate y(x), I get x × the original function y.1193

Let us do a verification.1227

Verify that the function y(x) = 5e ⁺x²/ 2 is a solution of this differential equation.1230

y’ is equal to 5e ⁺x²/ 2 × the derivative of that which is going to be 2x²/ 2 which is x.1256

y’ is equal to 5x e ⁺x²/ 2.1272

Now I substitute into the differential equation.1278

Put this into the differential equation and check to see.1290

y’ that is equal to 5x e ⁺x²/ 2, the question is does that equal x × y?1293

y was 5e ⁺x²/ 2, yes it does, that is a solution.1305

Let me go to blue, I think blue is my favorite color for these.1321

Instead of y(x) = 5 × e ⁺x²/ 2, I could have given you the following.1325

I could have given you y(x) = c × e ⁺x²/ 2, where c is any constant or c is any constant.1339

You can verify that this where c is any constant, it does not have to be the 5 will work.1365

Since c can be any number, this y(x) = c e ⁺x²/ 2, it represents an infinite family of functions, an infinite family of solutions.1374

We call it the general solution.1418

Once you have verified that, if you just use c instead of 5, it still satisfies the differential equation.1438

You are just going to get cx e ⁺x²/ 2 for our y’.1446

It satisfies it.1450

Basically, c could be any number.1452

Whatever c is, y could be 1 e ⁺x²/ 2, 5e ⁺x²/ 2, -15e ⁺x²/ 2, π e ⁺x²/ 2, it does not matter.1454

What it gives you is an infinite family of solutions.1468

We call that a general solution.1471

Anything that involves a constant, we call that the general solution.1472

This y(x) = 5e ⁺x²/ 2 is called a particular solution.1479

In differential equations, we speak of general solutions and we speak of particular solutions.1494

Notice that the thing that I want to show you in just a moment, that y (x) = c × e ⁺x²/ 2 is a family of curves.1503

Because this function y = is just a function of x, y = a function of x.1520

We know that y as a function of x is just a graph.1526

It is just a curve in the xy plane, that is it.1527

It is already a function, we can graph it.1530

It is a family of curves.1532

There you go, that is it.1543

A solution to a differential equation is also a curve in the xy plane, in the Cartesian coordinate system,1548

because it is just a function of x.1556

You are looking for some unknown function of x.1558

A function of x is a curve, that is it, it is all it is.1561

In this particular case, this curve right here, this is when c is equal to 2.1563

This curve represents when c is equal to 3.1572

This curve represents when c is equal to 4, and so on.1576

This curve right here, this was our y = 5e ⁺x²/ 2, that is it.1582

When a differential equation is given alone, it is the general solution that we are finding.1607

In other words, the one with constants.1631

In this particular case, y(x) = ce ⁺x²/ 2.1642

That is what this is, it is a family of solutions.1649

When a differential equation is given with a set of initial conditions, when a de is given with a set of initial conditions,1658

I will describe what those are in just a minute.1677

In other words, a certain function of y for a certain function of x that I know,1684

conditions such as y’ = xy, that is the differential equation and y(1.5) is equal to 7.2.1689

Now I do not know the function but I know an initial condition.1704

I know that at 1.5, when x = 1.5, I know the value of y is 7.2.1710

This is called an initial condition.1715

When a de is given with a set of initial conditions, in this case, just one initial condition, this is called an initial value problem.1720

In other words, you will also see it as just plain old ivp.1743

After we find the general solution, we use the initial condition to find a particular solution.1748

We use the initial condition to find, find a particular solution, to find a specific value for c.1784

Once we find the general solution, we use the initial condition next in the general solution, to find the specific value for c.1795

For y’ = xy, we found that y(x) is equal to some constant × e ⁺x²/ 2.1816

They tell us that, another thing that we know is that y(1.5) = 7.2.1833

Let us put 1.5 in for x.1838

y(1.5) which is equal to c × e¹.5²/ 2, they are telling me that it actually = 7.2.1841

I solve this equation for c.1854

c = 2.337, my particular solution in this case is y(x) is equal to 2.337 e ⁺x²/ 2.1863

This is my particular solution.1878

Find the general solution, use the initial conditions to find c.1883

That is this curve right here.1894

This curve is y = 2.337 e ⁺x²/ 2.1896

In this particular case, 1.5, 7.2, that is that point right there.1907

An initial condition is a point through which the particular solution passes, when you are looking at an actual curve of the solution.1919

From a family of solutions, we have reduced it to one, by use of an initial condition.1931

Let us see, let us move on to do some examples.1939

Again, in this lesson, we are only concerned with introducing you to differential equations,1945

having you do some basic verification, a little bit of manipulation.1949

In the following lessons, we will actually start working on how to solve these differential equations.1953

Verify that the family of functions y(x) = c/ x³ + 5x/ 4, is a solution of the differential equation xy’ + 3y = 5x.1959

How do we do a verification?1970

We find the derivatives, however many we need, first, second, third, whatever.1972

We put them in and we verify that left side is actually equal to the right side.1976

y(x) = that, let me work in blue.1984

I have got y(x) is equal to, I’m going to write it as cx⁻³ + 5/ 4x.1987

y’ is equal to -3x⁴ + 5/4.1998

Now I substitute into the de, substitute into the differential equation.2010

I have got x × y’ which I just found which is -3x⁻⁴ + 5/4 + 3y + 3 × y which is c/ x³ + 5x/ 4.2024

The question is does it equal 5x?2055

This is x × -3/ x⁴ + 5/4 + 3c/ x³ + 15x/ 42064

which = -3c/ x³ + 5x/ 4 + 3c/ x³ + 15x/ 4, 20x/ 4 5x.2084

Yes, this is a solution of the differential equation.2112

Notice this has a constant in it.2118

This is a family of solutions for this particular differential equations, with different values of c.2125

I have the different values of c over here.2131

In one of the cases, one c is equal to 1, I have the black curve.2134

That is this thing, that is this one.2138

That is a solution of the differential equation.2144

It is a graphical solution of the algebraic equation.2147

When y = 10, I have the orange curve.2150

Your orange curve is right there, and so on.2153

When c = 20, when c = -1, -10, -20, this is a family of curves.2159

For what values of k does the function y = k ⁺x satisfy the differential equation 3y” + 6y’ - 9 = 0.2170

For what values of k, interesting.2181

We have a function, we have the differential equation.2190

Let us just differentiate twice, plug it in and see what happens.2194

y is equal to e ⁺kx, y’ is equal to ke ⁺kx, and y” is equal to k² e ⁺kx.2202

Let us put these into the differential equation.2219

I have got 3 × k² e ⁺kx + 6 × y’ which is ke ⁺kx - 9 is equal to 0.2222

There is a little bit of mistake here, sorry about that.2254

Let me go to black, I forgot my y, I apologize for that.2257

This is supposed to be -9y is equal to 0.2265

Now let me go back to blue and finish this off.2274

-9 × y which is e ⁺kx = 0.2278

I have the function, I have the differential equations for what values of k I have differentiated.2288

I have this equation k.2293

Let me factor out the e ⁺kx.2295

I’m left with 3k² + 6k - 9 is equal to 0.2298

e ⁺kx is greater than 0 for all x.2309

Therefore, this is equal to 0.2314

3k² + 6k - 9 is equal to 0.2323

Let us go ahead and divide by 3 and make it a little easier on myself.2327

k² + 2x - 3 = 0, this one happens to factor, if not, no big deal.2330

We will just a graphing device or quadratic equation.2338

We have got k + 3 – 1.2345

We have k = -3, we have k = 1.2353

You have y is equal to e ⁻3x.2363

y is equal to e ⁺x.2370

When k is a -3 or 1, those two values of k satisfy this particular differential equation.2372

That is it, nice and straightforward.2383

Verify that the family of functions y(x) = c × e¹/ x + 7/ x + 7 is a solution to the differential equation x³ y’ + xy = 7.2388

Then, solve the initial value problem differential equation + y of 5 = 5.2400

Let us see what we can do.2409

We have the y = this thing.2410

Let us go ahead and find y’.2415

y’ = the derivative of this.2417

It is going to be c × e¹/ x × -1/ x² - 7/ x² + 0.2421

Now we go ahead and we substitute this into here.2438

We are going to write x³ × -c/ x² e¹/ x - 7/ x² + x × y + x × y2443

which is c × e¹/ x + 7/ x + 7.2465

The question is does that equal 7?2475

Here we are going to get –cx e¹/ x - 7/ x + cx e¹/ x.2480

What is going on, I’m losing my way here.2484

x × c ⁺7x/ 7.2516

Wait a minute, I have got all these symbols floating around.2527

We took the derivative, - c/ x² that is correct, - c/ x² e¹/ x.2532

This is -7/ x², that was our y’.2542

x³ × that, this is not 7/x, this is 7x x³.2547

There you go, + x × that, perfect.2562

+ 7 + 7x, sorry about that, it happens a lot.2572

Here we have – cx e¹/ x + cx e¹/ x - 7x + 7x.2579

Yes, we are left with 7.2587

It definitely equals that, great.2589

Now let us go ahead and solve the initial value problem.2593

y(5) = 5, we have y(x) = ce¹/ x.2598

We just verified that this is a solution, + 7/ x + 7.2612

They tell me that y(5) which is equal to ce¹/5 + 7/ 5 + 7, they are telling me that actually = 5.2620

When I solve for c, I get 2.78.2636

Therefore, our particular solution is y(x) is equal to - 2.78 e¹/ x + 7/ x + 7.2641

Now I have a function, the unknown function that I actually solve for a particular situation.2658

Now no matter what value of x I have, I can tell you what y is going to be.2665

This represents the family of solutions.2671

This is the family y(x) = ce¹/ x + 7/ x + 7.2676

That is all of these, the black.2697

When c is equal to 1, we have the black curve right here.2699

When c is equal to -1, we have the blue curve.2706

It looks a little different, that is here.2711

It comes up like that.2713

When y is equal to 10, we have the purple curve.2716

This one right here, that right there.2721

This is the family of curves.2724

Of course, the particular solution that we found.2730

This is y = - 2.78 e¹/ x + 7/ x + 7.2734

This is the particular solution to our initial value problem which was x³ y’ + xy = 7, y(5) = 5.2753

5,5, here is the point, 5,5 that is what this tells me.2770

The curve passes through that point.2776

I have an initial value.2780

The general solution, I have an initial value.2781

I can actually find a specific curve that satisfies this differential equation.2784

That is it for our introduction to differential equations, I hope that made sense.2791

Thank you so much for joining us here at www.educator.com.2794

We will see you next time, bye.2796

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