Differential Equations Answers

Let p be a real number. Consider the PDEs
xu_x + yu_y = pu −∞< x < ∞, −∞ < y < ∞.
(a) Find the characteristic curves for the equations.
(b) Let p = 4. Find an explicit solution that satisfies u = 1 on the circle x2 + y2 = 1.
(c) Let p = 2. Find two solutions that satisfy u(x, 0) = x^2, for every x > 0.

I know how to solve a) but b,c was not that easy. Solution for b should be u(x,y)=(x^2+y^2) and for c= u(x; y) = x^2 + ky^2, where k is a real nr.
u_x = partial of u with respect to x
u_y = partial of u with respect to y

Thanks for the help it's much appriciated!
Melvin

If u is a solution of the wave equation u_tt - c^2u_xx = 0 on -infinity < x < infinity, t > 0 for which u-->0, u_x-->0, and u_t-->0 as x-->+/-infinity, then the enrgy E = the integral from -infinity to +infinity of
(u^2_t + c^2u^2_x)dx is constant.
Is there a corresponding conserved quantity E* for solutions of the equation u_tt - c^2u_xx - bu = 0?

a) Solve the equation uu_x+u_y=1 with the initial condition u(x,x)=0. Hints: when finding the characteristic curves, solve for u before solving for x. When expressing t in terms of x and y, remember that t=0 on the initial curve.
b) Find the domain of the solution u(x,y) found in part a).

Suppose u is a solution of the wave equation u_tt - c^2u_xx = 0 on -infinity less than x less than 0, t greater than 0, and suppose that u-->0, u_x-->0, and u_t-->0 as x approaches +/- infinity. Show that E = ∫∞−∞ (u^2_t+c^2u^2_x)dx is constant.