Differential equation in a distribution space
I need your help to solve the following exercise:
find the general solution of the equation
x^2*u'(x) = pv(1/x)
where u'(x) is the derivative of u with respect to x and pv(1/x) stands for 'principal value of 1/x'.
The answer should be u(x) = c1 + c2*theta(x) + c3*delta(x) - 1/2*fp(1/x^2), where theta(x) is the Heaviside's function and fp(1/x^2) is the 'finite part of 1/x^2'.
For the sake of clarity, I solved similar equations in distributions spaces making use of Green's functions, Fourier transforms and convolutions, etc., but I can't figure out how to approach this one.
Thanks in advance for your suggestions!