Topic 3 of 6

Laplace transforms

Step 20 · Differential equations topics

Turn a differential equation into an algebra problem with the Laplace transform, solve it, and transform back using a table.

Transforming functions

The Laplace transform turns a function of time f(t) into a function F(s) by an integral. You rarely work out the integral: a short table covers most needs.

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

f(t)	F(s)
1	$\frac{1}{s}$
$t^{n}$	$\frac{n!}{s^{n+1}}$
$e^{at}$	$\frac{1}{s - a}$
$\sin bt$	$\frac{b}{s^{2} + b^{2}}$
$\cos bt$	$\frac{s}{s^{2} + b^{2}}$

Example: From the definition

Find $\mathcal{L}\{e^{2t}\}$ from the definition.

$\int_{0}^{\infty} e^{-st}e^{2t}\,dt = \int_{0}^{\infty} e^{-(s - 2)t}\,dt$ .
For s > 2 this is 1 ÷ (s − 2).

Answer $\frac{1}{s - 2}$

Example: Linearity

Find $\mathcal{L}\{3 + 5\sin 2t\}$ .

The transform of a sum is the sum of the transforms, and constants come out.
$\frac{3}{s} + 5 \cdot \frac{2}{s^{2} + 4}$ .

Answer $\frac{3}{s} + \frac{10}{s^{2} + 4}$

Transforming derivatives

Integrating by parts in the definition shows that a derivative becomes multiplication by s, minus the starting value. So derivatives turn into algebra, and the starting values come along automatically.

\mathcal{L}\{y'\} = sY - y(0) \qquad \mathcal{L}\{y''\} = s^{2}Y - sy(0) - y'(0)

Example: A first-order equation

Transform y′ + 3y = 0 with y(0) = 2, and solve.

sY − 2 + 3Y = 0.
So Y = 2 ÷ (s + 3).

Answer $Y = \frac{2}{s + 3}$ , so $y = 2e^{-3t}$

Solving with transforms

Transform both sides, using the starting values.
Solve the algebra for Y(s).
Split Y into table entries, often with partial fractions.
Transform back with the table.

Example: A second-order equation

Solve y″ − y = 0 with y(0) = 0 and y′(0) = 2.

Transform: $s^{2}Y - 2 - Y = 0$ , so $Y = \frac{2}{s^{2} - 1}$ .
Partial fractions: $\frac{2}{s^{2} - 1} = \frac{1}{s - 1} - \frac{1}{s + 1}$ .
Back with the table: $e^{t} - e^{-t}$ .

Answer $y = e^{t} - e^{-t}$

Example: With a force

Solve y′ + y = 1 with y(0) = 0.

Transform: $sY + Y = \frac{1}{s}$ , so $Y = \frac{1}{s(s + 1)}$ .
Split: $\frac{1}{s} - \frac{1}{s + 1}$ .

Answer $y = 1 - e^{-t}$

Try it yourself

Work each one on paper first, then open the answer to check.

Find $\mathcal{L}\{t^{2}\}$ .
Show answer
$\frac{2}{s^{3}}$
Find $\mathcal{L}\{e^{-4t}\}$ .
Show answer
$\frac{1}{s + 4}$
Find $\mathcal{L}\{\cos 3t\}$ .
Show answer
$\frac{s}{s^{2} + 9}$
Which function has transform $\frac{5}{s}$ ?
Show answer
The constant 5
Transform y′ − 2y = 0 with y(0) = 4, and solve.
Show answer
$y = 4e^{2t}$
Which function has transform $\frac{3}{s^{2} + 9}$ ?
Show answer
sin 3t