I have a PDE problem set due on monday and as I look at the teacher's method for solving this problem it doesn't make any sense to me. I'll show what I have so far.(adsbygoogle = window.adsbygoogle || []).push({});

Find the solution of

[tex] U^2U_x + U_y = 0 [/tex]

that satisfies [tex]u(x,0)=x[/tex]

So I started out with the characteristic equations the way my professor did it in his notes

[tex]x_t = u^2[/tex]

[tex]y_t = 1 [/tex]

[tex]z_t = 1[/tex]

then integrating these I get

[tex]t = \int 1/u^2 dx[/tex]

[tex]t = y + y_0 [/tex]

[tex]t = z + z_o[/tex]

I don't know how to integrate the u in the first equation (as its a function of x - and I don't think that integrating w.r.t t will help.) and using the intial conditions

[tex] x_o = s, y_o = 0, z_0 = s [/tex]

so that reduces them to

[tex]t = \int 1/u^2 dx[/tex]

[tex]t = y [/tex]

[tex]t = z + s[/tex]

and this is where I got stuck. I know I have to solve for x and y as functions of s and t and substitute them into the third equation to get the solution but I can't figure out how to do this while eliminating the s and t (I mean I could solve for s if I could do the integral in one but I can't.)

Anyways, all help would be greatly appreciated.

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# Partial Differential Equation Urgent Help

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