Unformatted text preview: ome arbitrary functions).
b) Find a particular solution up (x, y ), by using an approach similar to the
method of undetermined coeﬃcients (the “undetermined coeﬃcients” will
be function of x).
c) Form the general solution using the principle of superposition, and verify
that it is correct.
7. Consider the PDE (4) uyy + x2 uy = 0. a) Find the general solution.
b) Verify that your solution solves the DE.
c) Try to solve the boundary value problems consisting of the given equation
and the following boundary conditions. In each case state if there is a
unique solution, inﬁnitely many solutions, or no solution.
i) u(x, 0) = x2 ,
ii) u(x, 0) = x2 ,
iii) u(x, 0) = x2 , uy (0, y ) = y
uy (x, y ) y=x = ex
uy (1, y ) = e2y 8. Solve the PDE
2 u x + uy + u = 0
subject to the initial condition u(x, 0) = cos x.
9. Solve the initial value problem
3 y 2 ux
on the domain y 0. xuy = x, u(x, 0) = e x...
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 Fall '12
 DavidHamsworth
 Boundary value problem, general solution

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