Laplace Transforms

The Laplace transform converts a hard differential equation into a simple algebraic equation by mapping functions of time into the frequency domain.

Definition

$$\mathcal{L}\{f(t)\} = F(s) = \int_0^\infty e^{-st}f(t)\,dt$$

Where $t$ is time, $s$ is the complex transform variable, $f(t)$ is the input function, and $F(s)$ is its transform.

Essential Transform Pairs

Key Properties

• Linearity: $\mathcal{L}\{af+bg\} = aF + bG$
• First Derivative: $\mathcal{L}\{f'\} = sF(s) - f(0)$
• Second Derivative: $\mathcal{L}\{f''\} = s^{2}F(s) - sf(0) - f'(0)$

Solving with Laplace

Worked Example: Solve $y'' + 4y = 0$ with $y(0)=1$ and $y'(0)=0$.

1. Transform both sides: $(s^{2}Y - s - 0) + 4Y = 0$.
2. Solve algebraically for $Y$: $Y = \dfrac{s}{s^{2}+4}$.
3. Take the inverse transform: $y = \cos(2t)$.

Partial fractions are required to invert complex $F(s)$ expressions back into time-domain functions $f(t)$.

Physics Link: Essential for analyzing linear circuits and control systems, where initial conditions feed directly into the transformed equation.

Common Pitfall: Laplace transforms absorb initial conditions immediately via $\mathcal{L}\{y'\} = sY - y(0)$. Bolting them on at the end like standard ODEs will double-count or drop your values.