Laplace Transform
Convert differential equations to algebraic equations using the Laplace Transform integral formula.
The Formula
The Laplace Transform is a powerful integral transform that converts a function of time f(t) into a function of complex frequency F(s). Its greatest strength is that it transforms differential equations into algebraic equations, which are far easier to solve. Once you solve the algebraic equation in the s-domain, you apply the inverse Laplace Transform to get back to the time domain.
Introduced by Pierre-Simon Laplace in the late 18th century, this tool became essential in 20th-century engineering. It is the backbone of control theory, circuit analysis, signal processing, and mechanical vibration analysis. Without it, designing stable feedback systems, analyzing electrical circuits with capacitors and inductors, or solving spring-mass-damper problems would be vastly more difficult.
The key property that makes the Laplace Transform so useful is its effect on derivatives. The transform of f'(t) becomes sF(s) − f(0), and the transform of f''(t) becomes s2F(s) − sf(0) − f'(0). This means differentiation in the time domain becomes multiplication by s in the frequency domain. Initial conditions are automatically incorporated into the algebra.
Common Laplace Transform pairs are usually memorized or looked up in tables. For example, the transform of 1 is 1/s, the transform of eat is 1/(s−a), and the transform of sin(ωt) is ω/(s2+ω2). These pairs, combined with linearity and the derivative property, let you solve most standard differential equations in a few steps.
The variable s is generally a complex number (s = σ + jω), and the transform exists only when the integral converges. The region of convergence depends on the growth rate of f(t). For most practical engineering functions (exponentials, sinusoids, polynomials), the transform is well-defined for sufficiently large σ.
Variables
| Symbol | Meaning |
|---|---|
| F(s) | Laplace Transform of f(t) — a function in the s-domain |
| f(t) | Original time-domain function (defined for t ≥ 0) |
| s | Complex frequency variable (s = σ + jω) |
| t | Time variable (seconds) |
| e−st | Exponential kernel that weights the function |
Example 1
Find the Laplace Transform of f(t) = e3t.
F(s) = ∫0∞ e3t × e−st dt
F(s) = ∫0∞ e(3−s)t dt
F(s) = [e(3−s)t / (3−s)]0∞
For s > 3, the exponential goes to 0 as t → ∞
F(s) = 1 / (s − 3), valid for s > 3
Example 2
Solve y' + 2y = 6 with y(0) = 1 using the Laplace Transform.
Take the Laplace Transform of both sides: sY(s) − y(0) + 2Y(s) = 6/s
Substitute y(0) = 1: sY(s) − 1 + 2Y(s) = 6/s
Solve for Y(s): Y(s)(s + 2) = 6/s + 1 = (6 + s)/s
Y(s) = (s + 6) / (s(s + 2))
Partial fractions: Y(s) = 3/s − 2/(s + 2)
Inverse transform: y(t) = 3 − 2e−2t
When to Use It
The Laplace Transform is used whenever differential equations arise in engineering or physics.
- Analyzing and designing control systems (PID controllers, stability analysis)
- Solving circuit problems with capacitors, inductors, and resistors
- Modeling mechanical systems (vibrations, spring-mass-damper)
- Signal processing and filter design
- Solving initial value problems in ordinary differential equations