Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 17 / b / v / Solution

For a monomial trial $u_p=A{x}^m{y}^n$, the operator givesChoosing $m=n=2$ matches the two powers in the forcing, and $A=1/2$ matches their coefficients. A particular solution is therefore $u_p=x^2{y}^2/2$. Adding the homogeneous solutions givesThe arbitrary function labels the values on the characteristic curves; it must have sufficient regularity for the desired classical solution.