I came across a question where I had to apply Euler's theorem on a function U(x,y) which is represented in terms of f(y/x), g(y/x) and another expression in terms of x and y. The question is as follows:
If $$u(x,y) = \frac{(x^2+y^2)^n}{2n(2n-1)} + x f(\frac{y}{x})+g(\frac{y}{x}),$$ where f and g are arbitrary functions, prove by using Euler’s theorem that
$$\Bigl(x\frac{∂}{∂x}+y\frac{∂}{∂y}\Bigr)^2u(x,y)=(x^2+y^2)^n.$$
I wrote the first expressions as $$x^{2n}h(\frac{y}{x}),$$ expanded the square term, and tried to get the result, but in vain. I'm not sure if this is the right method and I messed up the calculation.
Any inputs on how to proceed would be really helpful. Kindly refrain from providing the actual answer. Thank you!
Edit: Another input would be that the sum of two homogeneous functions need not be homogeneous. How do we then apply Euler's Theorem?
Writing as an answer instead of a comment because use of pictures. Basically this is just a lot of algebra. Expand your operator as $$x^2\partial^2_x+xy\partial_x\partial_y+y^2\partial^2_y$$ And take the partial derivatives. Not bothered to do the algebra by hand, I checked using Mathematica, and the result is correct: