The adjoint of an equation in the form
So the adjoint for this DE is:
Any solution of the adjoint equation is an integrating factor for the original problem.
is one solution. Multiply through the original DE.
This is an exact equation, since
In the case of an exact equation the last relation equals 0, so
Use the integrating factor .
This equation is recognizable as an Euler equation since the exponent on the is the same as the order of the derivative in each term. So try for a solution in the form .
Substituting these in the equation gives:
The solution is therefore .