Find a second solution y2(x) of
x^2y"-3xy'+5y=0; y1=x^2cos(lnx)

We can try reduction order and look for a solution . Then

Substituting these into the ODE gives

which leaves us with an ODE linear in :

This ODE is separable; divide both sides by the coefficient of and separate the variables to get

Integrate both sides; on the right, substitute so that .

Now solve for ,

then for ,

Solve for by integrating both sides.

Substitute again and solve for :

then for ,

So the second solution would be

already accounts for the second term of the solution above, so we end up with

as the second independent solution.

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