I have to solve
$$9u_{xx} + 6u_{xy} + u_{yy} = 0 (*)$$
given $u(0, y) = cos(y)$ and $u_x(0, y) = y^2$
The function is parabolic so $\frac{dy}{dx} = \frac{1}{3}$ $\implies 3y - x$ is one parameter , $t$ and another is $x = s$.
Letting $u(x, y) = w(s(x,y). t(x,y))$, I derive the canonical form of the partial derivatives as follows:
$$u_{xx} = w_{ss} - 2w_{st} + w_{tt}$$ $$u_{xy} = 3w_{st} - 3w_{tt} $$ $$u_{yy} = 9w_{tt} $$
$$ \therefore (*) = 3w_{ss} + w_{st} - 4w_{tt} = 0$$
For the next step I need to integrate the last expression to find $w(s,t)$ so I can express it in terms of $u(x, y)$ and put in the initial conditions to solve, but I don't know how to integrate (*) so I am looking for advice please.
Thanks