# Canonical form of PDE (2nd order) and its general solution. [duplicate]

The PDE is given is: $$4u_{xx} + 5u_{xy} + u_{yy} + u_x + u_y = 2$$ I need to reduce this hyperbolic equation into a canonical form. I got the characteristics as $$\xi = y-x$$ and $$\eta = y - \frac{x}{4}$$
Let $$w(\xi,\eta) = u(x(\xi,\eta),y(\xi,\eta))$$. After substituting the partial derivatives of $$u$$ as pd's of $$w$$, I worked out the canonical form as (if anyone can confirm my answer it would be reassuring) $$w_{\xi\eta} - \frac{w_{\eta}}{3} = -\frac{8}{9}$$ How to solve this PDE to obtain the general solution? I proceeded in the following way: $$v = w_{\eta}$$, which gives $$v_{\eta} - \frac{v}{3} = -\frac{8}{9}$$ And solving this equation I got $$w(\xi,\eta) = \int 3e^{(\xi + g(\eta))/3} \ \partial \eta + \frac{8}{3}\eta + C$$ or $$u(x,y) = \boxed{ C + \frac{8}{3}\left(y-\frac{1}{4}x \right) + \int 3e^{(y-x + g(y-x/4))/3} \ (\partial y - \frac{1}{4}\partial x) }$$ I am not sure about the result. It looks messy and I don't know if my approach is correct. Any input is appreciated. Thanks in advance.