I know how to get general solution which is $$\phi(x,y)=F(x^2y)+G(xy^2)$$
now my question is if $$\phi(1,y)=10+y-2y^4$$ and $$\phi_x(1,y)=6+2y-4y^4$$ show that the unique solution is given by $$\phi(x,y)=10-2x^2y^4+yx^2+6lnx$$
General solution is not a problem.
Approach 1 : I couldn't get the particular solution directly from the initial conditions.
Approach 2 : I can prove the given solution indeed satisfies the PDE and the initial conditions but then i cant prove the uniqueness of it.
NOTE: Existence and uniqueness of PDE is not covered in the module.
I would be grateful for any help Thanks. couldn't find similar question on MSE