Find the general solution of the equation

3u_x+2u_y+(2x-3y)u=2xy-3y^2

The equation is solved as this example.

Example 2.7 Find the general solution to yur + Wy = x. In Example 2.6 we found the integral curves, so we set S = y2 - 2x. We also choose n(x, y) = y. The Jacobian determinant is J = 2 2y =-2 0 and is nonzero. Note that the inverse transformations are given by a = (n2 - 5)/2 and y = n. From these considerations, the partial differential equa- tion can be written as 12 - 5 un 2 Integrating this equation, we have U ( 8, n ) = 1 3 - En+ 9 ( 8 ) , for some differentiable function g. Hence, the general solution being sought is u(x, y) = =y3 - 54(32 - 2x) + 9(y2 - 2.)