| Title | Summary |
|---|---|
| Laplace Transform: The Definition | Paul's On-Line Notes defining the Laplace Transform. They use the definition to compute the Laplace transform of 1 and exp(at) as is done in our text. They also compute the Laplace transform of sin(at). Their approach requires two integration by parts. This is far more complicared than the approach taken in our text, which does not use any integration by parts. |
| Laplace Transform: Taking Transforms | Paul's On-Line Notes showing how to take the Laplace Transform using
linearity and a table. They do not show how the table is developed. |
| Laplace Transform: Inverse Transforms | Paul's On-Line Notes showing how to take the inverse Laplace Transform using
linearity, a table, and partial fraction identities. The approach to partial fractions is the slow approach you might have learned in your calculus class rather than the faster approach that we use and that is shown in the calculus review notes by Professor Levermore. |
| Laplace Transform: Step Functions | Paul's On-Line Notes showing how to use the unit step function to express a
list function in a form that allows it Laplace transform to be computed. The approach is not as systematic as the one in our notes. Examples taking the inverse Laplace Transform are not efficient because of the long way partial fraction identities are derived. |
| Laplace Transform: Solving Initial-Value Problems | Paul's On-Line Notes showing how to solve initial-value problems with the
Laplace transform. The method is illustrated with three examples. The derivation of partial fraction identities is not efficient. |
| Laplace Transform: Solving Initial-Value Problems with Step Functions | Paul's On-Line Notes showing how to use the Laplace transform to solve
initial-value problems with a forcing expressed in terms of step functions. The method is illustrated with four examples. |
| Introduction to the Laplace Transform | Professor Mattuck (MIT video) introducing the Laplace transform. |
| Derivative Formulas | Professor Mattuck (MIT video) deriving and applying the formulas for the Laplace transforms of derivatives of a function. |
| Convolution Formula | Professor Mattuck (MIT video) deriving and applying the formula for the Laplace transform the convolution of two functions. This material is in our text, but generally is not covered. |
| Using Laplace Transform to Solve ODEs with Discontinuous Inputs | Professor Mattuck (MIT video) solving a second-order nonhomogeneous linear
initial-value problem with a discontinuous forcing by the Laplace transform method. |
| Khan Academy: Laplace Transform 1 | Khan Academy video introducing the Laplace transform. |
| Khan Academy: Laplace Transform 2 | Khan Academy video introducing the Laplace transform. |
| Khan Academy: Transform of sin(at) | Khan Academy video computing the Laplace transform of sin(at). The approach requires two integration by parts and extends over two videos. This is far more complicared than the approach in our text, which does not use any integration by parts. |
| Khan Academy: Transform of sin(at) - Part 2 | Khan Academy video that is the continuation of the previous calculation. |
| Khan Academy: Laplace as Linear Operator and Laplace of Derivatives | Khan Academy video showing the linearity of the Laplace transform and the formula for the Laplace transform of a derivative. |
| Khan Academy: Laplace Transform of cos(t) and Polynomials | Khan Academy video using properties of the Laplace transform to derive formulas
for the transform of cos(at) and of powers t^n. The formula for the Laplace transform of t^n is worked out for n = 1,2,3, and seeing a pattern, a general formula is claimed that can be shown by induction, but no details are given. Given the formula for the Laplace transform of t^n and the linearity of the transform, the Laplace transform of any polynomial can be taken. |
| Khan Academy: Shifting a Transform by Multiplying by an Exponential | Khan Academy video working out further properties of the Laplace Transform. The transform of higher derivatives is shown. It also shows the property that the transform of e^{at} f(t) is F(s-a), where F(s) is the transform of f(t). |
| Khan Academy: Laplace Transform of the Unit Step Function | Khan Academy video computing the transform of the unit step functions. |
| Khan Academy: Inverse Laplace Examples | Khan Academy video showing by examples how to go from the Laplace transform Y(s) of a function back to the function y(t) --- taking an inverse Laplace transform. |
| Khan Academy: Laplace Transform to Solve an Equation | Khan Academy video showing how to use the Laplace transform to solve an initial-value problem with a homegeneous linear differential equation. (The title could be better.) Specifically, it shows how to compute the Laplace transform Y(s) of the solution. |
| Khan Academy: Laplace Transform Solves an Equation 2 | Khan Academy video showing how to use the Laplace transform to solve an initial-value problem. (Again, the title could be better.) Specifically, it shows how to go from Y(s) to the solution y(t). The approach to partial fraction identities is poor, making the solution more complicated than it needs to be. |
| Khan Academy: Using the Laplace Transform to Solve a Nonhomogeneous Equation | Khan Academy video showing how to use the Laplace transform to solve an initial-value problem with a nonhomegeneous linear differential equation. (Again, the title could be better.) The approach to partial fraction identities is poor, making the solution more complicated than it needs to be. |
| Khan Academy: Laplace/Step Function Differential Equation | Khan Academy video showing how to use the Laplace transform to solve an
initial-value problem with a nonhomegeneous linear differential equation that has a forcing written in terms of a shifted unit step function. (Again, the title could be better.) It does not show how to express a forcing given by a list function in terms of shifted unit step functions. The approach to partial fraction identities is poor, making the solution more complicated than it needs to be. |