The question I'm solving is as follows.

The differential equation given by,

satisfying the initial conditions y = xg(x) and u = f(x) with

a) f(x) = 2x, g(x)= 1 has no solution.

b) f(x) = 2x^{2}, g(x) = 1 has infinite number of solutions

c) f(x) = x^{3}, g(x) = x has a unique solution

d) f(x) = x^{4}, g(x) = x has a unique solution.

My initial line of approach was that the give equation was similar to the Euler's theorem for a homogenous function 'u' whose degree was 2. Since u = f(x), f(x) must be also homogenous function of degree 2, and the options had only one of them with degree 2.

Am i right in my conclusion? If so can someone guide me on concluding the no. of solutions as infinite?

If not, can someone help me get started?

P.S. I tried using LATEX but i kept getting an image saying that my tag was more than 25 characters. Hence I had to revert to attaching a .jpeg file.

Thanks