Separating the variables to solve differential equations is a familiar and simple method, but limited in it's usefulness because most equations are not separable. If however the equation is of the form
where
and
are both of the form
then separability can be achieved with the substitution
Proof: if
then
Simplification of the right hand side returns
Now x cancels throughout to give
and this equation is separable.
Example Use the substitutionto transform and solve the differential equation
and solve it subject to
at
Separating the variables gives
Now we can integrate:
when
Multiply by
to get