Consider the first order Quasi linear PDE given by $$uu_x+u_y=1$$
As usual
$$J=\begin{vmatrix} f'(s_o) & a(f(s_0),g(s_0),h(s_0))\\ g'(s_o) & b(f(s_0),g(s_0),h(s_0)) \end{vmatrix}$$ The characteristic equations associated here are:
$ \dfrac{dx}{dt}=u,\dfrac{dy}{dt}=1,\dfrac{du}{dt}=1$
Solve the Cauchy Problem associated with the following data:
a) $x=s,y=s,z=\dfrac{s}{2}, 0 \le s \le 1$
Here $$J=\begin{vmatrix} 1 & \dfrac{s}{2}\\ 1 & 1 \end{vmatrix}$$
For $J=0$, $s=2$ which can't happen and so solution exists.
So solving the equations along the Cauchy data , we get $$u=t+\dfrac{s}{2},y=t+s,x=\dfrac{t^2}{2}+\dfrac{st}{2}+s$$ $y-u=\dfrac{s}{2},t=2u-y,x=\dfrac{1}{2}t(y)+2y-2u=y(u-\dfrac{y}{2})+2y-2u$
Solving which I get $$u=\frac{x-2y+\frac{y^2}{2}}{y-2}$$
The only place where problem can happen is where $y=2$ and I think it can't happen because $s$ is bounded by $1$.
b) $x=\frac{s^2}{2},y=s,z=s, 0 \le s \le 1$
Here $$J=\begin{vmatrix} s & s\\ 1 & 1 \end{vmatrix}=0$$ Having known this I look at if $$\frac{f'}{a}=\frac{g'}{b}=\frac{h'}{c}$$ is satisfied or not and they turn out to be equal.
So I set out to find the characteristics and $$x=\frac{t^2}{2}+st+\frac{s^2}{2},y=t+s,u=t+s$$ and as it turns out there are two solutions to this one. One is $u=y$ and the other one is $u=\sqrt{2x}$
(c) $x=s^2,y=2s,z=s, 0 \le s \le 1$
The question here says that this particular data admits no solution to the Cauchy data. So I set out to find the solution expecting a problem somewhere. The Jacobian turned out to be $0$. The characteristic equations came out to be $$u=t+s,y=t+2s,x=\frac{t^2}{2}+st+s^2$$
Solving all these three equations yielded $$x=\frac{2u^2-2uy+y^2}{2}$$
This is a quadratic equation in $u$ and upon solving , I arrived at $$u=\frac{y+\sqrt{4x-y^2}}{2}$$ Ofcourse this is welldefined for $4x \ge y^2$ , which I believe is true from the given initial conditions. I even checked out that this satisfies the given differential equation.
d) In the fourth bit , the question asks to check the conditions in (c) just with the domain $0 \le s\le 3$ and says the Jacobian vanishes only on a subset.I am a bit perplexed here. The jacobian in my calculation should have to be $$J=\begin{vmatrix} 2s & s\\ 2 & 1 \end{vmatrix}=0$$
Thanks for the help!!
