Cauchy Problem: One-Parameter Family of Solution Curves
Explain in some detail, the reason for your answer:
I cannot use mathematical symbols. Thus, I will let * denote a partial derivative. For example, u*x means the partial derivative of us with respect to x. Furthermore, I will let u*x=p and u*y=q. Also, I will let ^ denote a power. For example, x^2 means x squared, and, I will let / denote division. Here is my problem:
The PDE is: (1+q^2)u-xp=0. I need to find the solution which passes contains the curve x^2=2u and y=0.
Now, I have found the initial curve to be (using the strip conditions):
x=a(t)=t y=b(t)=0 u=c(t)=t^2/2 p=d(t)=t q=e(t)=the square root of [2-t]
I have also found the characteristic system to be:
dx/ds=-x
dy/ds=2qu
du/ds=-px+2(q^2)u
dp/ds=p-p(1+q^2)
dq/ds=-q(1+q^2)
Important: I must regard this system as a set of ODE's. By solving this, I will find the one-parameter family of solution curves x(s,t), y(s,t), u(s,t), p(s,t), q(s,t). (Remember, when solving for the constants in each of the ODE's, we set s=0 and let the initial curve be the initial value.) Then I will have to solve for s and t in terms of x and y. Then I plug that back into u(s,t) to get my answer: u(x,y). My problem is: I cannot solve the characterisitic system! I can solve this to find a complete integral, but I cannot find the one-parameter family of solution curves.