Evans M. Harrell II and James V. Herod*
At Georgia Tech, this test took 50 minutes (calculators allowed but no notes).
Background. The operator to consider in this test is defined by
L(u) := x2 u''(x) - 2 u(x),
for 1 < x < 3. (It begins at 1 because the equation becomes singular when x=0.) Perhaps it is helpful to notice that L(x2) = L(1/x) = 0. You do not have to show this.
Impose boundary (initial) conditions that u(1) = u'(1) = 0 for L on this page.
Find the following:
1. L*(u) = _______________________________________________
with conditions on u: __________________________________
2. The Green function for L is
G(x,t) = _______________________________________________
3. The Green function for L* is
G#(x,t) = _______________________________________________
4. The solution of L(u) = x3 , u(1) = u'(1) = 0, is:
u(x) = _______________________________________________
FORMULA OR KEY FACT (FOR POSSIBLE PARTIAL CREDIT):_________
In problem 5, we consider different boundary conditions, but still
L(u) := x2 u''(x) - 2 u(x).
The boundary conditions are of the strange form:
9 u(3) = u(1)
9 u'(3) - u'(1) = 12 u(3)
You may take as given that L*(1) = 0. You do not have to show this, and you do not have to get involved in the strange boundary conditions.
5.
a) Give an example of a function f(x) for which
L(u) = f(x)
has a solution.
ANSWER: f(x) = ____________________________________________________
b) What specific differential equation must be satisfied by a function G(x,t) such that a solution to part a) is given by
u(x) = the integral from 1 to 3 of G(x,t) f(t) dt ?
ANSWER________________________________ = __________________________
It is not necessary to solve for G or for u. (But I will be suitably impressed if you do!)
FORMULA OR KEY FACT (FOR POSSIBLE PARTIAL CREDIT):_________
________________________________________________________________
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