How can one tell if a solution is existent or unique? For example:
$yu_y+uu_x=u-y$
$u(x,1)=x$
I've found the solution to be $u=x+1-y$, but have been told there are infinitely many solutions.
Is there a condition that must be satisfied for a unique solution?
Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:
$\dfrac{dy}{dt}=y$ , letting $y(0)=1$ , we have $y=e^t$
$\dfrac{du}{dt}=u-y=u-e^t$ , letting $u(0)=u_0$ , we have $u=(u_0-t)e^t=(u_0-\ln y)y$
$\dfrac{dx}{dt}=u=(u_0-t)e^t$ , we have $x=(u_0+1-t)e^t+f(u_0)=u+y+f\left(\dfrac{u}{y}+\ln y\right)$ , i.e. $\dfrac{u}{y}+\ln y=F(x-y-u)$
Consider the general solution $x=u+y+f\left(\dfrac{u}{y}+\ln y\right)$ :
$u(x,1)=x$ :
$f(x)=-1$
$\therefore x=u+y-1$ , i.e. $u=x-y+1$
But consider the general solution $\dfrac{u}{y}+\ln y=F(x-y-u)$ :
$u(x,1)=x$ :
$F(-1)=x$ , which is impossible.
$\therefore$ There is no solution.
Hence the PDE has the unique solution $u=x-y+1$ .