Differential equations may be transformed by a change of variables, making them simpler, and often easier to solve. Either the independent variable – usuallyoror the dependent variable, usuallymay be transformed, or both. The transformation must be chosen carefully, since not all transformations will make the equation simpler.
For example, suppose we have to solve the equation
We may make the transformationso that
Use of the chain rule gives
Removing the variablefrom this expression gives us the operator
Substituting these into the original equation gives
Simplification givesand dividing by the common factorgiveswhich can be easily solved to givein terms ofand in terms ofby substituting
Transforming the– variable is a little simpler. Suppose we have the equationSubstituteso that The equation becomeswhich becomeswhich simplifies toand then toby separating the variables.