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## Homework Statement

write down the Lagrange-Charpit eqs for

[tex] \frac{ \partial u}{ \partial x} \frac{ \partial u}{ \partial y} - y \frac{ \partial u}{ \partial x} - x \frac{ \partial u}{ \partial y}= 0 [/tex]

and use them to show [tex] \frac{ d^2 p}{ d p^2} = P [/tex]

assuming that u = x^2 when y=0 determine the characteristic curves (x(t),y(t))

## The Attempt at a Solution

so out eq gives pq-yp-xq = 0 so F(x,y,u,p,q) = pq-yp-xq

so

F_x = -q

F_y = -p

F_p = q-y

F_q = p-x

F_u = 0

so the char. eqs are

[tex] \frac{ dx}{ dt} [/tex] = q-y

[tex] \frac{ dy}{ dt} [/tex] = p-x

[tex] \frac{ dx}{ dt} [/tex] = qp

[tex] \frac{ dx}{ dt} [/tex] = q

[tex] \frac{ dx}{ dt} [/tex] = p

so [tex] \frac{ dx}{ dt} [/tex] = q-p but then [tex]{ d^2 p}{ d p^2} = 0 [/tex]??? what am i doing wrong, any ideas anyone?