Laplace Methods Summary

Differential Equations · Axiom Academy

Reviewing how Laplace transforms convert differential equations to algebra, making initial value problems with discontinuous forcing remarkably tractable. The Laplace transform turns a function of time into a function of s , and — critically — turns a linear ODE into an algebraic equation. The derivative property is the engine of the whole method: it bakes the initial conditions directly into the transformed equation, so there's no separate step for finding arbitrary constants. The shifting theorems ( and delaying f(t) by multiplying by e^ -as ) are what make the method so much more than a table lookup — they let one basic transform cover a whole family of forced and delayed problems. Unit step and impulse functions give a clean algebraic language for switches and shocks — no piecewise case-work required. The solution method is always the same four moves: transform, solve for Y(s) algebraically, decompose with partial fractions, invert back to y(t) . Core Concept The Transform & Basic Table The transform exists for any function of exponential order — one that doesn't outgrow Me^ at for some constants M,a . A short table of transforms (built directly from this integral) covers almost every function that shows up in practice. Linearity: — transforms respect linear combinations, so build complicated transforms from simple pieces. Watch out for: convergence needs s large enough — e.g. only holds for s>a . Core Concept The Derivative Property

This is the written version of the interactive lesson above. See the full Differential Equations course.