Question #25891

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

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

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