Laplace Transform

Definition

The Laplace transform of $f(t)$ is

$$F(s)=\mathcal{L}\left\{f(t)\right\}={\int }_0^{\infty }{e}^{-st}f(t)dt.$$

Intuition

The Laplace transform converts differentiation into multiplication by $s$, turning a linear ODE with initial conditions into an algebraic equation in $s$. Solving algebraically and inverting (using partial fractions and the table below) recovers the time-domain solution — including initial conditions — without having to integrate the ODE directly.

Formal Description

Linearity

$$\mathcal{L}\left\{c_1{f}_1(t)+c_2{f}_2(t)\right\}=c_1\mathcal{L}\left\{f_1\right\}+c_2\mathcal{L}\left\{f_2\right\}.$$

Transforms of derivatives (via integration by parts)

$$\mathcal{L}\left\{\dot{x}\right\}=sX(s)-x(0),\mathcal{L}\left\{\ddot{x}\right\}=s^{2}X(s)-sx(0)-\dot{x}(0).$$

Solving a constant-coefficient ODE

For $a\ddot{x}+b\dot{x}+cx=g(t)$ with $x(0)=x_0$, $\dot{x}(0)=u_0$, the Laplace transform yields the algebraic equation

$$a\left(s^{2}X-s{x}_0-u_0\right)+b\left(sX-x_0\right)+cX=G(s).$$

Solve for $X(s)$, then invert using the table below (with partial fractions as needed) to obtain $x(t)$.

Table of Laplace Transforms

#	$f(t)={\mathcal{L}}^{-1}\{F(s)\}$	$F(s)=\mathcal{L}\{f(t)\}$
1	$e^{at}f(t)$	$F(s-a)$
2	$1$	$\frac{1}{s}$
3	$e^{at}$	$\frac{1}{s-a}$
4	$t^n$	$\frac{n!}{s^{n+1}}$
5	$t^n{e}^{at}$	$\frac{n!}{(s-a{)}^{n+1}}$
6a	$\sin bt$	$\frac{b}{s^2+b^2}$
6b	$\sinh bt$	$\frac{b}{s^2-b^2}$
7a	$\cos bt$	$\frac{s}{s^2+b^2}$
7b	$\cosh bt$	$\frac{s}{s^2-b^2}$
8	$e^{at}\sin bt$	$\frac{b}{(s-a{)}^2+b^2}$
9	$e^{at}\cos bt$	$\frac{s-a}{(s-a{)}^2+b^2}$
10	$t\sin bt$	$\frac{2bs}{(s^2+b^2{)}^2}$
11	$t\cos bt$	$\frac{s^2-b^2}{(s^2+b^2{)}^2}$
12	$u_c(t)$	$\frac{e^{-cs}}{s}$
13	$u_c(t)f(t-c)$	$e^{-cs}F(s)$
14	$\delta (t-c)$	$e^{-cs}$
15	$\dot{x}(t)$	$sX(s)-x(0)$
16	$\ddot{x}(t)$	$s^{2}X(s)-sx(0)-\dot{x}(0)$

Applications

• Solving inhomogeneous ODEs with discontinuous or impulsive forcing (Heaviside, Dirac delta).
• Analysing stability and frequency response in control systems (transfer functions).
• Computing convolutions via the convolution theorem $\mathcal{L}\{f\ast g\}=F(s)G(s)$.

Trade-offs

• Most powerful when $g(t)$ involves Heaviside or Dirac delta terms; for smooth $g(t)$, undetermined coefficients may be simpler.
• Inverting $X(s)$ requires partial fraction decomposition and matching against the table; complex poles can make this tedious.
• The standard table assumes zero-state initial conditions are encoded in the transform; non-zero initial conditions add extra $s$-domain terms that must be tracked carefully.

Links

• Inhomogeneous Second-Order ODE
• Heaviside and Dirac Functions