Study GuideDifferential Equations–The Laplace Transform1. Solving Differential EquationsTheLaplace transformis a powerful tool for solvingfirst-and second-order differential equationswith constant coefficients, especially when the problem is aninitial value problem(IVP).A key requirement is that theinitial conditions are given at(x = 0).1.1Why Use Laplace Transforms?Normally, when solving an IVP:1.You first find thegeneral solutionof the differential equation.2.Then you use the initial conditions to determine the constants.The Laplace transform works differently—and often more efficiently:•It automatically builds theinitial conditionsinto the solution process.•It turns adifferential equationinto analgebraic equation, which is usually easier to solve.1.2The Basic StrategyTo solve an IVP using Laplace transforms, follow these steps:1.Apply the Laplace transform(L) toboth sidesof the differential equation.2.Use properties of the Laplace transform to rewrite the equation in terms of(L[y]).3.Solve the resultingalgebraic equationfor(L[y]).4.Take theinverse Laplace transformto find(y(x)).1.3Laplace Transforms of DerivativesBefore applying the method, we need to know how the Laplace transform acts on derivatives.First derivativeUsing integration by parts, we find:Preview Mode
This document has 27 pages. Sign in to access the full document!