PDE Characteristic Curve Method

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Homework Statement


Solve [tex] u_x^2+u_y^2=1 [/tex] subject to [tex] u(x, ax)=1 [/tex]


Homework Equations





The Attempt at a Solution


I let [tex] u_x=p [/tex] and[tex]u_y=q [/tex]and[tex] F=p^2 +q^2 -1=0 [/tex] Then x'=2p, y'=2q, u'=p.2p+q.2q=2, and p'=0=q'. So [tex] p=p_0, q=q_0 [/tex] are constants. I got [tex] x'=2p_0, y'=2q_0 [/tex] and integrating the equations for x',y',u' I get [tex] x=2p_0s+x_0[/tex] [tex] y=2q_0s + y_0 [/tex]and [tex] u=2s+u_0 [/tex] I'm not sure when it comes to implementing the initial condition. I think that I can say that [tex] u_x(x,ax)=p_0=0[/tex] which gives q=+1 or -1 from the condition of F=0. Can I then fill in [tex] x_0, ax_0, 1 [/tex] for the initial values of x, y and u respectively? In the end I get an answer [tex] u=\frac{y}{q_0}-\frac{ax}{q_0}+1 [/tex]. Firstly, I'm not sure what to do when [tex] q_0[/tex] can have two values. Also, when I fill in my answer into the original PDE I get that a must be zero, which can't be right.

Thanks for any help.
 

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