Differential equations may be transformed by a change of variables, making them simpler, and often easier to solve. Either the independent variable – usually
or
or the dependent variable, usually
may 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 transformation
so that
Use of the chain rule gives
Removing the variable
from this expression gives us the operator
Then
Substituting these into the original equation gives
Simplification gives
and dividing by the common factor
gives
which can be easily solved to givein terms of
and in terms ofby substituting
Transforming the– variable is a little simpler. Suppose we have the equation
Substitute
so that
The equation becomes
which becomes
which simplifies to
and then to
by separating the variables.