While solving a Goursat problem I stumbled upon this PDE. Can't solve it: $$8v_{\xi\eta} - v_\xi - v_\eta = 0$$
This is what I tried:
$v \rightarrow V e^{\alpha x + \beta x}$
$v_\xi = V_\xi e^{\alpha x + \beta x} + \alpha V e^{\alpha x + \beta x}$
$v_\xi = V_\eta e^{\alpha x + \beta x} + \beta V e^{\alpha x + \beta x}$
$v_{\xi\eta} = V_{\xi\eta} e^{\alpha x + \beta x} +\beta V_\xi e^{\alpha x + \beta x} + \alpha V_\eta e^{\alpha x + \beta x} + \alpha \beta V e^{\alpha x + \beta x}$
Replacing these in the equation and simplifying by $e^{\alpha x + \beta x}$ I got:
$$ V_{\xi\eta} + (\beta -1) V_\xi + (\alpha - 1) V_\eta + \alpha\beta V - \alpha V - \beta V = 0 $$
And we get a not solvable system: $ \begin{cases} \alpha - 1 = 0 \\ \alpha \beta - \alpha - \beta = 0 \end{cases} $, so I can't get rid of $V_\xi$, $V$ or $V_\eta$, $V$.
