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## Homework Statement

Consider the PDE [itex]xu_x + y u_y = 4 u, -\infty < x < \infty, -\infty < y < \infty[/itex]. Find an explicit solution that satisfies [itex]u = 1[/itex] on the ellipse [itex]4x^2 + y^2 = 1[/itex].

## Homework Equations

## The Attempt at a Solution

The characteristic curves are

[itex]x(t,s) = f_1(s) e^t[/itex]

[itex]y(t,s) = f_2(s) e^t[/itex]

[itex]u(t,s) = f_3(s) e^{4t}[/itex].

The initial conditions are

[itex]x(0,s) = s[/itex]

[itex]y(0,s) = \pm \sqrt{1 - 4s^2}[/itex]

[itex]u(0,s) = 1[/itex].

Parametric representation of the integral surface is then

[itex]x(t,s) = s e^t[/itex]

[itex]y(t,s) = \pm \sqrt{1 - 4s^2} e^t[/itex]

[itex]u(t,s) = e^{4t}[/itex].

How do I invert these to get [itex]u(x,y)[/itex]?