Consider the equation
xy'' - y' + 4x^(3)y = 0
show that y1(x) = sin(x^2) is a solution
find the second solution by letting y = u(x)y1 (x) and solve for u(x)
write the complete general solution

Substituting and its derivatives into the ODE gives

as required.

Assume a second solution of the form . Then

Substitute these into the ODE:

This ODE is separable as

Integrating both sides gives

For the remaining integral, we have

So we have

Integrate both sides to solve for :

Then our second solution turns out to be

already captures the second term here, so the second fundamental solution is

and the general solution is