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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.

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marked as duplicate by Harry49, YuiTo Cheng, cmk, metamorphy, Badam Baplan Jun 27 at 16:38

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