Anyone can help me? Please. This is for my thesis.
How to derive the solution of $$u.u_{xy} - u_{x}u_{y} - u_{y} = 0$$ with known $u(0,y)$, $u(0,x)$ and $u(y,y) = 0$
You can state the solution in the form of $u(0,y)$ and $u(0,x)$
The solution is $$u(x,y) = \frac{u(0,y) - u(0,x)}{\frac{\partial u(0,x)}{\partial x}}$$
First we rewrite the equation as $$ u^2(u_{xy}u^{-1}-u_xu_yu^{-2}-u_yu^{-2})=u^2(u_xu^{-1}+u^{-1})_y=0 $$ Thus we can integrate with respect to $y$ and get: $$ u_x(x,y)+1=u(x,y)c(x) $$ For an arbitrary function $c(x)$. The solution to this ODE can be found by the usual standard methods and reads: $$ u(x,y)=\frac{f(y)-a(x)}{a'(x)} $$ For $c(x)=-(\ln(a'(x))'$. Inserting the initial conditions yields the solution.