Homework Statement
Consider the PDE xu_x + y u_y = 4 u, -\infty < x < \infty, -\infty < y < \infty. Find an explicit solution that satisfies u = 1 on the ellipse 4x^2 + y^2 = 1.
Homework Equations
The Attempt at a Solution
The characteristic curves are
x(t,s) = f_1(s) e^t...
I tried this and it turns out to be the same as what I got before.
\int^t_0 \frac{1}{\sqrt{s^2 + r}} dr = \int^{s^2+t}_{s^2} \frac{1}{\sqrt{v}} dv = \left[2 v^{\frac{1}{2}} \right]^{s^2+t}_{s^2} = 2\sqrt{s^2 + t} - 2s
y(t) = 2\sqrt{s^2 + t} - s
Then
s = 2x - y
t = x^2 - (2x - y)^2
and
u =...
Why does \frac{dy}{dt} = \frac{1}{\sqrt{s^2 + t}} imply y(t) - s = \int^t_0 \frac{1}{\sqrt{s^2 + u}} du?
How do you get the -s on the LHS of the second equation?
Why is the integration on the RHS only from 0 to t? (I just realized that s > 0, is this related to the integral?)
What is...
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Solve the following Cauchy problem
\displaystyle \frac{1}{2x}u_x + xu u_y + u^2 = 0,
subject to
\displaystyle u(x,x) = \frac{1}{x^2}, x >...
This question is also posted at http://www.mathhelpforum.com/math-help/f59/similarity-solutions-185537.html. Please view that post instead for better formatting.
The original question is:
Try and apply the Similarity solution method to the following boundary value problems for u(x,t).
u_t...