Why is initial curve not assumed to be coincident with characteristics I was reading the method of characteristic for first order p.d.e's.
$a\frac{\partial u}{\partial x}+b\frac{\partial u}{\partial y}=c$.
Now it said to find out the solution we try to find out a specific curve along which the pde becomes an ode. This part I get, since the curve is where the first order derivatives get discontinuous. But why does it say that the initial data curve should not coincide with this chatacteristic. What will happen if the initial conditions are specified on the boundary. Can somebody give an example.?
Say i have the equation as $$yp+q=2$$ with $u$ known on the initial curve $y=0, 0\leq x\leq1$. So i get the characteristics and solution as
$y^2=2(x-x_R)$ and $u= 2y +u_R$

If you give data along a characteristic curve, the solution $u$ has two possibly conflicting jobs to do: (1) satisfy the ODE along the characteristic, and (2) satisfy the given condition along the characteristic.
As a simple example, try solving the PDE $u_x=0$ subject to the condition $u(x,0)=x$. Good luck! :-)
