particular integral of PDE

Question

Let $P(x,y)$ be a particular integral of the partial differential equation $$z_{xx} -z_y= 2y -x^2$$ Then $P(2,3)$ equals (a) 2 (b) 8 (c) 12 (d) 10

Answer 1

Trying a 3rd degree polynomial solution $$z=a+b x+c y+d x y+e x^2+f y^2+g x^3+h x^2 y+i x y^2+j y^3 $$ gives a 4-parameter family of solutions $$a+b x+e x^2+2 e y+g x^3+6 g x y+x^2 y$$ which could have any value at the point $(2,3)$.