General Contents

Ordinary Differential Equations and Laplace Transforms

The links below are grouped by topic. They are labeled with icons:

To solutions in Aid for Calculus

To solutions in Paul's Online Math Notes

To solutions in Cliff's Notes

Separable Differential Equations,
Homogeneous Differential Equations>,
Exact Differential Equations,
Integrating Factors,
Linear Differential Equations,
Linear Homogeneous Differential Equations (Second Order),
Bernoulli Differential Equations,
Linear Nonhomogeneous DE, Undetermined Coefficients,
Linear Nonhomogeneous DE, Variation of Parameters
Reduction of Order
Laplace Transforms
Inverse Laplace Transforms
Initial Value Problems and Laplace Transforms

If you don't find a problem type you are looking for in this list, please recommend what I should add. Email to John Taylor

Separable Differential Equations

Separable: Example1: separable differential equation

separable differential equation

Separable: Example2: separable differential equation

separable differential equation

Separable: Example3: separable differential equation

separable differential equation

Separable: Example4: separable differential equation

separable differential equation

Separable: Example5: separable differential equation

separable differential equation

Separable: Example6: separable differential equation and initial value

separable differential equation and initial value

Separable Khan Video 1 Khan eqn 1

Khan eqn 1

Separable Khan Video 2 Khan eqn 2

Khan eqn 2

Separable Paul Example 1

Paul example 1

Separable Paul Example 2

Paul example 2

Separable Paul Example 3

Paul example 3

Separable Paul Example 4

Paul example 4

Separable Paul Example 5

Paul example 5

Separable Paul Example 6

Paul example 6

Scroll to Example 1 Cliff example 2

Cliff example 2

Scroll to Example 2 Cliff example 2

Cliff example 2

Scroll to Example 3 Cliff example 3

Cliff example 3

Scroll to Example 4 Cliff example 4

Cliff example 4

Scroll to Example 5 Cliff example 5

Cliff example 5

Scroll to Example 6 Cliff example 6

Cliff example 6

Scroll to Example 7 Cliff example 7

Cliff example 7

Homogeneous Differential Equations

Homogeneous: Example1: Homogeneous differential equation

Homogeneous differential equation

Homogeneous: Example2: Homogeneous differential equation

Homogeneous differential equation

Homogeneous: Example3: Homogeneous differential equation

Homogeneous differential equation

Homogeneous Paul Example 1 Paul Homogeneous 1

Paul Homogeneous 1

Homogeneous Paul Example 2 Paul Homogeneous 2

Paul Homogeneous 2

Scroll to Example 1 Cliff example 1

Cliff example 1

Scroll to Example 2 Cliff example 2

Cliff example 2

Scroll to Example 3 Cliff example 3

Cliff example 3

Scroll to Example 4 Cliff example 4

Cliff example 4

Scroll to Example 5 Cliff example 5

Cliff example 5

Scroll to Example 6, 7 Cliff example 6, 7

Cliff example 6, 7

Scroll to Example 8 Cliff example 8

Cliff example 8

Exact Differential Equations

Exact: Example 1: Exact differential equation

Exact differential equation

Exact: Example 2: Exact differential equation

Exact differential equation

Exact: Example 3: Exact differential equation

Exact differential equation

Exact Paul Example 1 Paul Exact 1

Paul Exact 1

Exact Paul Example 2 Paul Exact 2

Paul Exact 2

Exact Paul Example 3 Paul Exact 3

Paul Exact 3

Exact Paul Example 4 Paul Exact 4

Paul Exact 4

Exact Paul Example 5 Paul Exact 5

Paul Exact 5

Scroll to Example 1 Cliff Exact 1

Cliff Exact 1

Scroll to Example 2 Cliff Exact 2

Cliff Exact 2

Scroll to Example 3 Cliff Exact 3

Cliff Exact 3

Scroll to Example 4 Cliff Exact 3

Cliff Exact 3

Scroll to Example 5 Cliff Exact 3

Cliff Exact 3

Scroll to Example 6 Cliff Exact 3

Cliff Exact 3

Integrating Factors

Integrating Factor Example 1: differential equation requiring an integrating factor

differential equation requiring an integrating factor

Integrating Factor Example 2: differential equation requiring an integrating factor

differential equation requiring an integrating factor

Integrating Factor Example 3: differential equation requiring an integrating factor

differential equation requiring an integrating factor

Integrating Factor Example 4: differential equation requiring an integrating factor

differential equation requiring an integrating factor

Integrating Factor Example 5: differential equation requiring an integrating factor

differential equation requiring an integrating factor

Integrating Factor Example 6: differential equation requiring an integrating factor

differential equation requiring an integrating factor

Paul Example 1 Paul Integrating Factor 1

Paul Integrating Factor 1

Paul Example 2 Paul Integrating Factor 2

Paul Integrating Factor 2

Paul Example 3 Paul Integrating Factor 3

Paul Integrating Factor 3

Paul Example 4 Paul Integrating Factor 4

Paul Integrating Factor 4

Paul Example 5 Paul Integrating Factor 5

Paul Integrating Factor 5

Paul Example 6 Paul Integrating Factor 6

Paul Integrating Factor 6

Scroll to Example 1 Cliff Integrating Factor 1

Cliff Integrating Factor 1

Scroll to Example 2 Cliff Integrating Factor 2

Cliff Integrating Factor 2

Scroll to Example 3 Cliff Integrating Factor 3

Cliff Integrating Factor 3

Scroll to Example 4 Cliff Integrating Factor 4

Cliff Integrating Factor 4

Linear Differential Equations

Linear Differential Equations: Example 1:

linear differential equation

Linear Differential Equations: Example 2:

linear differential equation

Paul Example 1 Paul Linear Differential Equations 1

Paul Linear Differential Equations 1

Paul Example 2 Paul Linear Differential Equations 2

Paul Linear Differential Equations 2

Paul Example 3 Paul Linear Differential Equations 3

Paul Linear Differential Equations 3

Paul Example 4 Paul Linear Differential Equations 4

Paul Linear Differential Equations 4

Paul Example 5 Paul Linear Differential Equations 5

Paul Linear Differential Equations 5

Paul Example 6 Paul Linear Differential Equations 6

Paul Linear Differential Equations 6

Scroll to Example 1 Cliff Linear Differential Equations 1

Cliff Linear Differential Equations 1

Scroll to Example 2 Cliff Linear Differential Equations 2

Cliff Linear Differential Equations 2

Scroll to Example 3 Cliff Linear Differential Equations 3

Cliff Linear Differential Equations 3

Scroll to Example 4 a. Cliff Linear Differential Equations 4

Cliff Linear Differential Equations 4

b.

Cliff Linear Differential Equations 4

Scroll to Example 5 Cliff Linear Differential Equations 5

Cliff Linear Differential Equations 5

Scroll to Example 6 Cliff Linear Differential Equations 6

Cliff Linear Differential Equations 6

Linear Homogeneous Differential Equations (Second Order)

Concepts

Paul 1

Paul Concepts of linear homogeneous 2nd order equations 1

Paul 2

Paul Concepts of linear homogeneous 2nd order equations 2

Paul 3

Paul Concepts of linear homogeneous 2nd order equations 3

Real, Distinct Roots

Linear Homogeneous Differential Equations: Example 1:

Linear homogeneous differential equation (second order)

Linear Homogeneous Differential Equations: Initial Conditions: Example 1: Linear homogeneous ordinary differential equation

Linear homogeneous ordinary differential equation

Paul 1

Paul Real, Distinct Roots of linear homogeneous 2nd order equations 1

Paul 2

Paul Real, Distinct Roots of linear homogeneous 2nd order equations 2

Paul 3

Paul Real, Distinct Roots of linear homogeneous 2nd order equations 3

Paul 4

Paul Real, Distinct Roots of linear homogeneous 2nd order equations 4

Paul 5

Paul Real, Distinct Roots of linear homogeneous 2nd order equations 5

Scroll to Example 1 Cliff Real, Distinct Roots of linear homogeneous 2nd order equations 1

Cliff Real, Distinct Roots of linear homogeneous 2nd order equations 1

Scroll to Example 2 Cliff Real, Distinct Roots of linear homogeneous 2nd order equations 2

Cliff Real, Distinct Roots of linear homogeneous 2nd order equations 2

Scroll to Example 5 Cliff Real, Distinct Roots of linear homogeneous 2nd order equations 5

Cliff Real, Distinct Roots of linear homogeneous 2nd order equations 5

Complex Roots

Linear Homogeneous Differential Equations: Example 3: Linear homogeneous ordinary differential equation

Linear homogeneous ordinary differential equation

Linear Homogeneous Differential Equations: Boundary Values: Example 1:

Linear homogeneous differential equation (second order), initial condition

Paul 1

Paul Complex Roots of linear homogeneous 2nd order equations 1

Paul 2

Paul Complex Roots of linear homogeneous 2nd order equations 2

Paul 3

Paul Complex Roots of linear homogeneous 2nd order equations 3

Paul 4

Paul Complex Roots of linear homogeneous 2nd order equations 4

Scroll to Example 4 Cliff Repeated Roots of linear homogeneous 2nd order equations 4

Cliff Repeated Roots of linear homogeneous 2nd order equations 4

Repeated Roots

Linear Homogeneous Differential Equations: Example 2:

Linear homogeneous differential equation (second order)

Linear Homogeneous Differential Equations: Initial Conditions: Example 2: Linear homogeneous ordinary differential equation

Linear homogeneous ordinary differential equation

Paul 1

Paul Repeated Roots of linear homogeneous 2nd order equations 1

Paul 2

Paul Repeated Roots of linear homogeneous 2nd order equations 2

Paul 3

Paul Repeated Roots of linear homogeneous 2nd order equations 3

Scroll to Example 3 Cliff Repeated Roots of linear homogeneous 2nd order equations 3

Cliff Repeated Roots of linear homogeneous 2nd order equations 3

Bernoulli Differential Equations

Bernoulli Differential Equations: Example 1:

Bernoulli differential equation

Bernoulli Differential Equations: Example 2:

Bernoulli differential equation

Bernoulli Differential Equations: Example 3:

Bernoulli differential equation

Bernoulli Differential Equations: Example 4:

Bernoulli differential equation

Bernoulli Differential Equations: Example 5:

Bernoulli differential equation

Bernoulli Differential Equations: Example 6:

Bernoulli differential equation

Paul Example 1 Paul Bernoulli 1

Paul Bernoulli 1

Paul Example 2 Paul Bernoulli 2

Paul Bernoulli 2

Paul Example 3 Paul Bernoulli 3

Paul Bernoulli 3

Paul Example 4 Paul Bernoulli 4

Paul Bernoulli 4

Scroll to Example 1 Cliff Bernoulli 1

Cliff Bernoulli 1

Linear Nonhomogeneous Differential Equations: Undetermined Coefficients

Linear Nonhomogeneous Differential Equations Undetermined Coefficients: Example 1:

Nonhomogeneous differential equation requiring undetermined coefficients and involving x

Linear Nonhomogeneous Differential Equations Undetermined Coefficients: Example 2:

Nonhomogeneous differential equation requiring undetermined coefficients and involving a polynomial

Linear Nonhomogeneous Differential Equations Undetermined Coefficients: Example 3:

Nonhomogeneous differential equation requiring undetermined coefficients and involving an exponential

Linear Nonhomogeneous Differential Equations Undetermined Coefficients: Example 4:

Nonhomogeneous differential equation requiring undetermined coefficients and involving a cosine

Linear Nonhomogeneous Differential Equations Undetermined Coefficients: Example 5:

Nonhomogeneous differential equation requiring undetermined coefficients and involving a cosine

Linear Nonhomogeneous Differential Equations Undetermined Coefficients: Example 6:

Nonhomogeneous differential equation requiring undetermined coefficients and involving an exponential

Linear Nonhomogeneous Differential Equations Undetermined Coefficients: Example 7:

Nonhomogeneous differential equation requiring undetermined coefficients and involving the product of a cosine and an exponential

Linear Nonhomogeneous Differential Equations Undetermined Coefficients: Example 8:

Nonhomogeneous differential equation requiring undetermined coefficients and involving the product of an exponential and x

Paul 1

Paul Undetermined Coefficients 1

Paul 2

Paul Undetermined Coefficients 2

Paul 3

Paul Undetermined Coefficients 3

Paul 4

Paul Undetermined Coefficients 4

Paul 5

Paul Undetermined Coefficients 5

Paul 6a Right-hand side = Paul Undetermined Coefficients 6a

Paul Undetermined Coefficients 6a

Paul 6b Right-hand side = Paul Undetermined Coefficients 6b

Paul Undetermined Coefficients 6b

Paul 6c Right-hand side = Paul Undetermined Coefficients 6c

Paul Undetermined Coefficients 6c

Paul 7

Paul Undetermined Coefficients 7

Paul 8a Right-hand side = Paul Undetermined Coefficients 8a

Paul Undetermined Coefficients 8a

Paul 8b Right-hand side = Paul Undetermined Coefficients 8b

Paul Undetermined Coefficients 8b

Paul 8c Right-hand side = Paul Undetermined Coefficients 8c

Paul Undetermined Coefficients 8c

Paul 8d Right-hand side = Paul Undetermined Coefficients 8d

Paul Undetermined Coefficients 8d

Paul 8e Right-hand side = Paul Undetermined Coefficients 8e

Paul Undetermined Coefficients 8e

Paul 8f Right-hand side = Paul Undetermined Coefficients 8f

Paul Undetermined Coefficients 8f

Paul 8g Right-hand side = Paul Undetermined Coefficients 8g

Paul Undetermined Coefficients 8g

Paul 9

Paul Undetermined Coefficients 9

Paul 10a

Paul Undetermined Coefficients 10a

Paul 10b

Paul Undetermined Coefficients 10b

Paul 10c

Paul Undetermined Coefficients 10c

Paul 10d

Paul Undetermined Coefficients 10d

Paul 10e

Paul Undetermined Coefficients 10e

Scroll to Example 3 Cliff Undetermined Coefficients 3

Cliff Undetermined Coefficients 3

Scroll to Example 4 Cliff Undetermined Coefficients 4

Cliff Undetermined Coefficients 4

Scroll to Example 5 Cliff Undetermined Coefficients 5

Cliff Undetermined Coefficients 5

Scroll to Example 6 Cliff Undetermined Coefficients 6

Cliff Undetermined Coefficients 6

Scroll to Example 7 Cliff Undetermined Coefficients 7

Cliff Undetermined Coefficients 7

Scroll to Example 8 Cliff Undetermined Coefficients 8

Cliff Undetermined Coefficients 8

Scroll to Example 9 Cliff Undetermined Coefficients 9

Cliff Undetermined Coefficients 9

Linear Nonhomogeneous Differential Equations: Variation of Parameters

Linear Nonhomogeneous Differential Equations Variation of Parameters: Example 1: Nonhomogeneous linear differential equation requiring variation of parameters

Nonhomogeneous linear differential equation requiring variation of parameters

Linear Nonhomogeneous Differential Equations Variation of Parameters: Example 2: Nonhomogeneous linear differential equation requiring variation of parameters

Nonhomogeneous linear differential equation requiring variation of parameters

Linear Nonhomogeneous Differential Equations Variation of Parameters: Example 3: Nonhomogeneous linear differential equation requiring variation of parameters

Nonhomogeneous linear differential equation requiring variation of parameters

Linear Nonhomogeneous Differential Equations Variation of Parameters: Example 4: Nonhomogeneous linear differential equation requiring variation of parameters: third derivative

Nonhomogeneous linear differential equation requiring variation of parameters: third derivative

Linear Nonhomogeneous Differential Equations Variation of Parameters: Example 5: Nonhomogeneous linear differential equation requiring variation of parameters

Nonhomogeneous linear differential equation requiring variation of parameters

Linear Nonhomogeneous Differential Equations Variation of Parameters: Example 6: Nonhomogeneous linear differential equation requiring variation of parameters

Nonhomogeneous linear differential equation requiring variation of parameters

Paul 1

Paul Variation of Parameters 1

Paul 2

Paul Variation of Parameters 2

Paul 3

Paul Variation of Parameters 3

Scroll to Example 1 Cliff Variation of Parameters 1

Cliff Variation of Parameters 1

Scroll to Example 2 Cliff Variation of Parameters 2

Cliff Variation of Parameters 2

Scroll to Example 3 Cliff Variation of Parameters 3

Cliff Variation of Parameters 3

Linear Differential Equations: Reduction Of Order

One nonzero solution is known:

Example 1: Aid4Calc Reduction of Order 1

Aid4Calc Reduction of Order 1

Example 2: Aid4Calc Reduction of Order 2

Aid4Calc Reduction of Order 2

Example 3: Aid4Calc Reduction of Order 3

Aid4Calc Reduction of Order 3

Paul 1

Paul Reduction of Order 1

Paul 2

Paul Reduction of Order 2

Scroll to Example 5 Cliff Reduction of Order 5

Cliff Reduction of Order 5

Scroll to Example 6 Cliff Reduction of Order 6

Cliff Reduction of Order 6

With no x factor or term:

Example 1: Linear differential equation requiring reduction of order

Linear differential equation requiring reduction of order

Example 2: Linear differential equation requiring reduction of order

Linear differential equation requiring reduction of order

Scroll to Example 4 Cliff Reduction of Order 4

Cliff Reduction of Order 4

With no y factor or term:

Example 1: Linear differential equation requiring reduction of order

Linear differential equation requiring reduction of order

Example 2: Linear differential equation requiring reduction of order

Linear differential equation requiring reduction of order

Scroll to Example 1 Cliff Reduction of Order 1

Cliff Reduction of Order 1

Scroll to Example 2 Cliff Reduction of Order 2

Cliff Reduction of Order 2

Scroll to Example 3 Cliff Reduction of Order 3

Cliff Reduction of Order 3

Laplace Transform

1: Find the Laplace transform of Laplace Transform of powers and exponentials

Laplace Transform of powers and exponentials

2: Find the Laplace transform of Laplace Transform of sines and cosines

Laplace Transform of sines and cosines

3: Find the Laplace transform of Laplace Transform of hyperbolic sines and cosines

Laplace Transform of hyperbolic sines and cosines

4: Find the Laplace transform of Laplace Transform of product of exponential and cosine

Laplace Transform of product of exponential and cosine

5: Find the Laplace transform of Laplace Transform of product of t and a hyperbolic cosine

Laplace Transform of product of t and a hyperbolic cosine

6: Find the Laplace transform of Laplace Transform of product of t squared and cosine

Laplace Transform of product of t squared and cosine

7: Find the Laplace transform of Laplace Transform of a half integer power

Laplace Transform of a half integer power

8: Find the Laplace transform of Laplace Transform of a scaled half integer power

Laplace Transform of a scaled half integer power

Paul 1a

Paul Laplace Transform 1a

Paul 1b

Paul Laplace Transform 1b

Paul 1c

Paul Laplace Transform 1c

Paul 1d

Paul Laplace Transform 1d

Paul 2a

Paul Laplace Transform 2a

Paul 2b

Paul Laplace Transform 2b

Paul 2c

Paul Laplace Transform 2c

Paul 2d

Paul Laplace Transform 2d

Paul 2e

Paul Laplace Transform 2e

Scroll to Example 1 Cliff Laplace_Transform 1

Cliff Laplace_Transform 1

Scroll to Example 2 Cliff Laplace_Transform 2

Cliff Laplace_Transform 2

Scroll to Example 3 Cliff Laplace_Transform 3

Cliff Laplace_Transform 3

Scroll to Example 4 Cliff Laplace_Transform 4

Cliff Laplace_Transform 4

Scroll to Example 5 Cliff Laplace_Transform 15

Cliff Laplace_Transform 15

Scroll to Example 6 Cliff Laplace_Transform 6

Cliff Laplace_Transform 6

Scroll to Example 7 Cliff Laplace_Transform 7

Cliff Laplace_Transform 7

Scroll to Example 8 Cliff Laplace_Transform 8

Cliff Laplace_Transform 8

Inverse Laplace Transform

1a: Find the Inverse Laplace transform of Inverse Laplace Transform of inverse linear

Inverse Laplace Transform of inverse linear

1b: Find the Inverse Laplace transform of Inverse Laplace Transform of inverse linear and power

Inverse Laplace Transform of inverse linear and power

1c: Find the Inverse Laplace transform of Inverse Laplace Transform, sum of squares in denominator

Inverse Laplace Transform, sum of squares in denominator

1d: Find the Inverse Laplace transform of Inverse Laplace Transform, sum and difference of squares in denominator

Inverse Laplace Transform, sum and difference of squares in denominator

2a: Find the Inverse Laplace transform of Inverse Laplace Transform, sum of squares in denominator

Inverse Laplace Transform, sum of squares in denominator

2b: Find the Inverse Laplace transform of Inverse Laplace Transform, non-factorable quadratic in denominator

Inverse Laplace Transform, non-factorable quadratic in denominator

2d: Find the Inverse Laplace transform of Inverse Laplace Transform, factorable quadratic in denominator

Inverse Laplace Transform, factorable quadratic in denominator

3a: Find the Inverse Laplace transform of Inverse Laplace Transform, factored cubic in denominator

Inverse Laplace Transform, factored cubic in denominator

3b: Find the Inverse Laplace transform of Inverse Laplace Transform, a linear and a difference of squares in denominator

Inverse Laplace Transform, a linear and a difference of squares in denominator

Paul 1a

Paul Inverse Laplace Transform 1a

Paul 1b

Paul Inverse Laplace Transform 1b

Paul 1c

Paul Inverse Laplace Transform 1c

Paul 1d

Paul Inverse Laplace Transform 1d

Paul 2a

Paul Inverse Laplace Transform 2a

Paul 2b

Paul Inverse Laplace Transform 2b

Paul 2c

Paul Inverse Laplace Transform 2c

Paul 2d

Paul Inverse Laplace Transform 2d

Paul 3a

Paul Inverse Laplace Transform 3a

Paul 3b

Paul Inverse Laplace Transform 3b

Paul 3c

Paul Inverse Laplace Transform 3c

Scroll to Example 9 Cliff Inverse Laplace_Transform 9

Cliff Inverse Laplace_Transform 9

Scroll to Example 10 Cliff Inverse Laplace_Transform 10

Cliff Inverse Laplace_Transform 10

Scroll to Example 11 Cliff Inverse Laplace_Transform 11

Cliff Inverse Laplace_Transform 11

Scroll to Example 12 Cliff Inverse Laplace_Transform 12

Cliff Inverse Laplace_Transform 12

Initial Value Problems and Laplace Transform

Differential equation for initial value problem via Laplace Transforms

Differential equation for initial value problem via Laplace Transforms

If you don't find a problem type you are looking for in this list, please recommend what I should add. Email to John Taylor