Advanced Engineering Mathematics, 7th Edition

2.5 Solutions by Substitutions

INTRODUCTION

We usually solve a differential equation by recognizing it as a certain kind of equation (say, separable) and then carrying out a procedure, consisting of equation-specific mathematical steps, that yields a function that satisfies the equation. Often the first step in solving a given differential equation consists of transforming it into another differential equation by means of a substitution. For example, suppose we wish to transform the first-order equation dy/dx = f(x, y) by the substitution y = g(x, u), where u is regarded as a function of the variable x.

If g possesses first-partial derivatives, then the Chain Rule gives

By replacing dy/dx by f(x, y) and y by g(x, u) in the foregoing derivative, ...