Convert differential equations into algebra — then transform back.
The Laplace transform converts time-domain functions to s-domain using an improper integral. The exponential kernel e⁻ˢᵗ ensures convergence for suitable s.
The derivative property is the key insight: differentiation becomes multiplication by s. This turns second-order DEs into algebraic equations in s — much easier to solve!
Transform: s²Y − s − 0 + 3(sY − 1) + 2Y = 0
(s² + 3s + 2)Y = s + 3 → Y = (s + 3)/((s + 1)(s + 2))
Partial fractions: Y = 2/(s + 1) − 1/(s + 2)
Inverse: y(t) = 2e⁻ᵗ − e⁻²ᵗ
The workflow: (1) transform the DE, (2) solve the algebraic equation for Y(s), (3) use partial fractions and the table to invert.
Step functions model sudden switches (turning on a force). The delta function models instantaneous impulses (a hammer strike). These are essential in engineering and signal processing.