MATH 20D Lecture 21

Chapter 7 - Laplace Transform

Section 7.2 & 7.3 - Definition and Properties of Laplace Transform

Laplace Transform

Given a function defined for all , then the Laplace Transform of is given by

Reason to use Laplace Transform
ODE Soln to given ODE is hard
ODE Algebraic Equation Soln to Algebraic Equation Soln to Given ODE is easier
Going from Soln to the Algebraic Equation to Soln to Given ODE requires an Inverse Laplace Transform

Fact

Let . If is piecewise continuous and for “big enough” then the Laplace transform exists for all .
In addition, Laplace transform is a linear operator: for any functions :

provided that each transformation exists.

Example. Let $f(t)=e^{at}$. Find $\mathcal{L}\{f\}$.

If (or equivalently, ), then limit D.N.E. As a result, Laplace transform also D.N.E.
If (or ) then this limit is 0, and thus, the transform is now
Therefore: whenever

Example. Let $f(t)=1$. Find $\mathcal{L}\{f\}$.

If then limit D.N.E.
If then limit = 0 making
, for
A “smarter way” to derive this formula is to let in the previous formula.

Example. Let $f(t)=t$. Find $\mathcal{L}\{f\}$.

When , limit & Laplace transform D.N.E.
When , limit 0 making
So , for

Euler’s formula:
If :

:

Example. Let $f(t)=\sin at$ and $g(t)=\cos at$. Find $\mathcal{L}\{f\}$ and $\mathcal{L}\{g\}$.

, for

for
, for