# Non-homogenenous PDE: method of characteristics

I am very new here after a friend recommended this site. I am preparing for my final graduation exam there is one question I can not solve:

I was hoping perhaps someone could help me.

$$a= a(x,y)$$ $$a a_{xy} = a_x a_y = xy$$ $$a(x,0) = x$$ (so $a$ differentiated w.r.t. $x$ and $y$).

I need to find the solution of a using the method of characteristics. I only knew how to solve it using direct integration but this was not allowed