Properties of the Laplace Transform

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 7.3 – Properties of the Laplace Transform

Expected Educational Results

• Objective 20–1: I understand the properties of the Laplace transform.
• Objective 20–2: I can utilize the properties of the Laplace transform.

Properties of the Laplace Transform

Laplace Transform of the Derivative

Theorem

Let $f(t)$ be continuous on $[0,\infty)$ and of exponential order $\alpha$. Let $f^{\,\prime}(t)$ be piecewise continuous on $[0,\infty)$ and of exponential order $\alpha$. Then for $s>\alpha$, $\displaystyle \mathcal{L}\left\{f^{\,\prime}(t)\right\}(s)=s\mathcal{L}\left\{f(t)\right\}(s)-f(0)$

Investigation 04

Prove the above theorem.

Laplace Transform of the Higher Order Derivatives

Theorem

Let $f(t)$, $f^{\,\prime}(t)$, $f^{\,\prime\prime}(t)$, $\ldots$,$f^{\,(n-1)}(t)$ be continuous on $[0,\infty)$, let $f^{\,(n)}(t)$ be piecewise continuous on $[0,\infty)$, with all functions of exponential order $\alpha$. Then for $s>\alpha$,

$\displaystyle \mathcal{L}\left\{f^{\,(n)}(t)\right\}(s)=s^n\mathcal{L}\left\{f(t)\right\}(s)-s^{n-1}f(0)-s^{n-2}f^{\,\prime}(0)-s^{n-3}f^{\,\prime\prime}(0)-\cdots-f^{\,(n-1)}(0)$

NOTE: The Method of Mathematical Induction is required to prove the above theorem.

Essentially, the Laplace transform changes a differential equation in terms of $t$ into an algebraic expression in terms of $s$ which will be helpful in solving higher-order IVPs.

Theorem

Let $F(s)=\mathcal{L}\left\{f(t)\right\}(s)$ and assume $f(t)$ is piecewise continuous on $[0,\infty)$ and of exponential order $\alpha$. Then for $s>\alpha$, $\displaystyle \mathcal{L}\left\{t^nf(t)\right\}(s)=(-1)^n\frac{d^n}{ds^n}\mathcal{L}\left\{f(t)\right\}(s)$

Investigation 05

1. Evaluate $\displaystyle \mathcal{L}\left\{\sin{(t)}\right\}(s)$.
2. Evaluate $\displaystyle \mathcal{L}\left\{t\sin{(t)}\right\}(s)$.
3. Evaluate $\displaystyle \mathcal{L}\left\{t^2\sin{(t)}\right\}(s)$.
4. Evaluate $\displaystyle \mathcal{L}\left\{t^3\sin{(t)}\right\}(s)$.