MATH 519 Homework
Fall 2013
1. Let L = aD2 + bD + c (a 6= 0) be a constant coefficient second order linear differential
operator, and let p(λ) = aλ2 +bλ+c be the associated characteristic polynomial. If λ1 , λ2
are the roots of p, show that we can express the operator L as L = a(D − λ1 )(D − λ2 ).
Use this factorization to obtain the general solution of Lu = 0 in the case of repeated
roots, λ1 = λ2 .
√
2. Show that the solution of the initial value problem y 0 = 3 y, y(0) = 0 is not unique.
(Hint: y(t) = 0 is one solution, find another one.) Why doesn’t this contradict the
general theorem about unique solvability of the initial value problem?
3. For λ 6= 0 show that the unique solution of
Z x
u(y) dy = λu(x) + g(x)
0
may be written as
g(x)
1
u(x) = −
− 2
λ
λ
Z
x
e
x−y
λ
g(y) dy
0
4. Consider the integral equation
Z 1
K(x, y)u(y) dy = λu(x) + g(x)
0
for the kernel
K(x, y) =
x2
1 + y3
a) For what values of λ ∈ C does there exist a unique solution for any function
g ∈ C([0, 1])?
b) Find the solution set of the equation for all λ ∈ C and g ∈ C([0, 1]).
(Hint: For λ 6= 0 any solution must have the form u(x) = − g(x)
+ Cx2 for some constant
λ
C. We are using the usual notation that g ∈ C([0, 1]) if g is continuous on [0, 1].)
R1
5. Find a kernel K(x, y) such that u(x) = 0 K(x, y)f (y) dy is the solution of
u00 + u = f (x)
u(0) = u0 (0) = 0
(Hint: Review the variation of parameters method in any undergraduate ODE textbook.)