partial differential equation

Jul 2009
The question I'm solving is as follows.

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) = 2x2, g(x) = 1 has infinite number of solutions
c) f(x) = x3, g(x) = x has a unique solution
d) f(x) = x4, 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.