A table of transforms · Differential Equations

Intuition

A handful of transforms, computed once, solve almost every problem of this chapter: powers of t, exponentials, sines and cosines, and the rule that multiplying by an exponential shifts the transform. Each entry is one integration, usually by parts. After that the table is used the way a multiplication table is: forwards to transform, and backwards to invert.

A phrasebook. Working out each sentence from the grammar every time would be slow; a few memorised phrases, and the rule for combining them, cover almost every conversation.

The pairs this chapter uses

The transforms below hold for $s$ large enough: $s>0$ , or $s>a$ where an exponential $e^{at}$ appears. Read from right to left, the table inverts a transform. The inverse transform $\mathcal{L}^{-1}$ is well defined, because two continuous functions with the same transform are equal — Lerch's theorem, used here without proof.

\begin{aligned}\mathcal{L}\{t^{n}\}&=\frac{n!}{s^{n+1}}\\\mathcal{L}\{\sin\omega t\}&=\frac{\omega}{s^{2}+\omega^{2}}\\\mathcal{L}\{\cos\omega t\}&=\frac{s}{s^{2}+\omega^{2}}\end{aligned}

Using the table

The shift rule: $\mathcal{L}\{e^{at}f(t)\}=F(s-a)$ . So $L {e^{2 t} \cos 3 t$ .

The shift rule

\mathcal{L}\{e^{at}f(t)\}(s)=F(s-a)

Write out the transform of the product. The two exponentials combine into one with rate s minus a, and what is left is exactly the transform of f, evaluated at s minus a instead of at s.

Proof steps

\mathcal{L}\{e^{at}f(t)\}=\int_{0}^{\infty}e^{-st}e^{at}f(t)\,dt

Write out the definition.

=\int_{0}^{\infty}e^{-(s-a)t}f(t)\,dt

Combine the exponentials.

=F(s-a)

This is the transform of f with s − a in place of s.

Applications

Practice

Sines and Cosines

The sine has ω on top and the cosine has s. Both share the denominator s² + ω².

\begin{aligned}\mathcal{L}\{\sin\omega t\}&=\frac{\omega}{s^{2}+\omega^{2}}\\\mathcal{L}\{\cos\omega t\}&=\frac{s}{s^{2}+\omega^{2}}\end{aligned}

Try it

What is $\mathcal{L}\{\sin3t\}$ ?