425/525, Math Spring 2011
Jerry L. Kazdan
Problem Set 1
D UE : In class Thursday, Jan. 27. Late papers will be accepted until 1:00 PM Friday.
1. Find Greens function g(x, s) to get a formula u(x) =
of u (x) = f (x) .
Rx
0
g(x, s) f (s) ds for a particular solution
2. In class we considered the oscillations of a weight attached to a spring hanging from the ceiling.
If u(t ) is the displacement of the mass m we were let to solve mu(t ) = ku , where k > 0 is
a constant that depends on the stiffness of the spring. But this model neglected gravity. If we
include gravity the equation becomes
mu = ku + mg,
where g is the gravitational constant,
Solve this equation assuming you know the initial conditions u(0) = A and u (0) = B .
3. Let a(x) and f (x) be periodic functions with period P , so, for instance, a(x + P) = a(x) . This
problem investigates periodic solutions u(x) (with period P ) of Lu := u (x) + a(x)u = f (x) .
a) Show there is a periodic solution of u (x) = f (x) if and only if
RP
0
f (x) dx = 0 .
b) Show that the homogeneous equation Lu = 0 has a non-trivial P -periodic solution u(x) if
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and only if 0P a(x) dx = 0 .
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c) If 0P a(x) dx = 0 , show that the inhomogeneous equation Lu = f always has a unique
P -periodic solution u(x) .
R
On the other hand, if 0P a(x) dx = 0 , nd a necessary and sufcient condition for Lu = f
to have a P -periodic solution. If it has P a periodic solution, is this solution unique?
4. In class we obtained a simpler general formula for a particular solution of the inhomogeneous
rst order system U + AU = F , where U (x) and F (x) are vectors with n components and
A(x) is an n n matrix. In addition we showed how much of the theory for a second order
equation was in fact a special case of that for a rst order system.
Use this to re-derive Lagranges formula for a particular solution of the inhomogeneous equation u + u = f .
5. Show that the boundary value problem u + u = f R on 0 x with boundary conditions
u(0) = 0 and u() = 0 has a solution if and only if 0 f (x) sin x dx = 0 .
Bonus Problems (Due Jan. 27)
1-B Let a(t ) and f (t ) be periodic continuous functions with period 2 .
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a) Show that the equation u = f has a 2 -periodic solution (so both u and u are 2 periodic) if and only if
Z 2
0
f (t ) dt = 0.
b) Show that the equation u + u = f has a 2 -periodic solution if and only if both
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0 and 02 f (t ) cos t dt = 0 .
R 2
0
f (t ) sin t dt =
c) More generally, show that the equation Lu := u + a(t )u = f has a 2 -periodic solution if
R
and only if 02 f (t )z(t ) dt = 0 for all 2 -periodic solutions of z + a(t )z = 0 . [R EMARK :
These are special cases of the Fredholm alternative: the image of L is the orthogonal
complement of the kernel of the adjoint operator L .]
[Last revised: January 25, 2011]
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