Math 401 final exam april 13, 2012. Total: 50 pts: consider the ode problem for u(x): (cid:26) u + u 2u = f (x), u (0) + u(0) = 0, 0 < x < 1 u(1) = 2 (1) (a) (4 pts. ) Write down the problem satis ed by the green"s function gx(y) = G(x; y) for problem (1) (but do not try to solve it). (b) (2 pts. ) Express the solution u(x) of (1) in terms of gx(y). (c) (4 pts. ) Now suppose the bc at x = 1 is changed to u(1) + au (1) = 2. For what value of a is a solvability condition on f (x) required for (1). 2: fix a number q > 0, and consider the operator + q in the plane r2. For this problem it may be helpful to recall the formula for the laplacian in polar coordinates = 2. Write down the equation that the free-space green"s function.
