What is the function that solves the Cauchy Problem?

Question

Solve the Cauchy Problem $u_x+(x+y) u_y=1, u(x,-x)=0$ using the method of characteristics.

I arrived the $c_2=u-x$ and $c_1=e^{(-x)}(y+x+1)$. Then $c_2=G(c_1)$, and using the initial conditions I get $G(e^{-x})=-x$. But after that I do not know how to get the function u that solves the problem.

Answer 1

Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:

$\dfrac{dx}{dt}=1$ , letting $x(0)=0$ , we have $x=t$

$\dfrac{dy}{dt}=x+y=t+y$ , we have $y=y_0e^t-t-1=y_0e^x-x-1$

$\dfrac{du}{dt}=1$ , letting $u(0)=f(y_0)$ , we have $u(x,y)=t+f(y_0)=x+f((x+y+1)e^{-x})$

$u(x,-x)=0$ :

$f(e^{-x})=-x$

$f(x)=\ln x$

$\therefore u(x,y)=x+\ln((x+y+1)e^{-x})=\ln(x+y+1)$