I need to solve Cauchy problem for partial differential equation. My equation looks like this: $$ 3\tfrac{\partial^2 u}{\partial x^2} + 5\tfrac{\partial^2 u}{\partial x\partial y} - 2\tfrac{\partial^2 u}{\partial y^2} + 7(\tfrac{\partial u}{\partial x}+2\tfrac{\partial u}{\partial y}) = 0. $$ and conditions look like: $u|_{x=0} = 1$, $\frac{\partial u}{\partial x}|_{x=0} = 3y$
My attemps: First of all, I know, to solve it, we need to transform this view to canonical view, I've done this, result is here : $$\dfrac {\partial^2 u}{\partial \eta \partial \xi} -\dfrac { \partial u}{\partial \eta}=0$$ And as I checked, this view corresponds to the canonical view of the hyperbolic equation (my equation is hyperbolic as D > 0)
and the replacement I did to transform initial view to canonical view is here: (so far as I know it is needed for the further solution Cauchy problem) $$\phi (x,y) = y-2x=c $$ $$\psi (x,y) = 3y+x=c$$ $$\eta= 3y+x $$ $$\xi = y-2x $$
Unfortunately, I don't know what I need to do next, I've been surfing internet and textbooks for hours, but haven't found any good explanations or just can't understand I've checked similar questions on math.stackexchange, but there is no something that is relevant for my topic
I am trying to do my best, but I need some help It seems I need to use integrals, do some actions with them, integrate, substitute and other stuff like that, as minimal textbooks are speaking about it, but can't find a clue for the solution
If I am right, I found function u(x,y): $$ u(x,y)=e^{y-2x}\phi_1(3y+x)+\phi_2(y-2x) $$ Now I need to use initial conditions to find $$ \phi_1, \phi_2 $$ I firstly differentiated u(x,y) and got: $$\dfrac { \partial u}{\partial x} = -2e^{y-2x}\phi_1(3y+x)+ \phi_1'(3y+x)e^{y-2x} -2\phi_1(y-2x) $$ Then using the initial conditions I got a system:
$$ \begin{cases} \begin{align} e^{y}\phi_1(3y) + \phi_2(y) &= 1 \\ -2e^{y}\phi_1(3y) + \phi_1'(3y)e^y - 2\phi_2'(y) &= 3y \end{align} \end{cases} $$
But now I am not succeed in getting functions $$ \phi_1, \phi_2 $$ can't find actions required to solve this system
Starting from your last line: $$\dfrac {\partial^2 u}{\partial \eta \partial \zeta} -\dfrac { \partial u}{\partial \eta}=0$$ $$\dfrac {\partial w}{\partial \zeta} -w=0$$
Where $w=\dfrac { \partial u}{\partial \eta}$.
Solve the equation : $$\dfrac {d w}{d \zeta} -w=0$$ $$ \dfrac {d }{d \zeta}(we^{-\zeta})=0$$ Integrate: $$we^{-\zeta}=C(\eta)$$ $$w=C(\eta)e^{\zeta}$$
Now you can solve the equation: $$w=\dfrac { \partial u}{\partial \eta}$$ $$\dfrac { \partial u}{\partial \eta}=C(\eta)e^{\zeta}$$ Finally: $$\boxed {u(\zeta,\eta)=\phi_1(\eta)e^{\zeta}+\phi_2(\zeta)}$$