MATHEMATICS 110/E-210, SPRING 2014 VECTOR SPACE

MATHEMATICS 110/E-210, SPRING 2014
VECTOR SPACE METHODS FOR DIFFERENTIAL EQUATIONS
Proof list for quiz 1
Last modified: January 25, 2014
On the first quiz on February 19, two of these proofs will appear: one chosen
at random from 1-4 and one from 5-7.
You can receive one point of extra credit for creating a pdf file of one of these
proofs and upoading it to the dropbox on the quiz 1 page before 10 PM Monday,
Feb. 17 for posting on the Web site. This offer becomes void for a given proof
once three files have been posted.
1. Using variation of parameters, show that the general solution to the firstorder initial value problem
l(y) = a(x)y 0 (x) + b(x)y(x) = h(x); y(x0 ) = 0
can be written as the definite integral
Z
x
y(x) = f (x)
x0
h(t)
dt
f (t)a(t)
where f (x) is an element of the kernel of l, and confirm this formula by
differentiation.
2. Define the Wronskian W (x) for n functions f1 (x), · · · fn (x), and prove that
if W (x) is nonzero for any x, the functions are linearly independent.
3. Start with the second-order equation l(y) = a(x)y 00 + b(x)y 0 + c(x)y = 0.
Show that the Wronskian W (x) of any two independent solutions f1 (x) and
f2 (x) of this equation satisfies the first-order equation
a(x)W 0 + b(x)W = 0 and that the solution of this equation is
Z
W = C exp(−
b(x)
dx).
a(x)
4. Prove that the Laplace transform of y 00 (x) is y˜00 (s) = s2 y˜(s) − sy(0) − y 0 (0).
1
P∞
00
0
i
5. Start with
the
equation
y
=
p(x)y
+
q(x)y,
where
p(x)
=
i=0 pi x , and
P∞
q(x) = i=0 qi xi .
P∞
j
Show that, in a solution of the form y(x) =
j=0 yj x , the coefficients
satisfy
(n + 2)(n + 1)yn+2 =
n+1
X
jpn+1−j yj +
n
X
j=1
qn−j yj .
j=0
You do not need to show convergence of the series!
6. Prove that the equation
Y 00 (x) =
N
M
0
Y
xY +
1− S
(1 − Sx )2
has a solution of the form
Y (x) = A(1 −
x −λ
)
S
with λ > 0.
Outline the strategy (details not required) for using this equation to prove
that the power series solution to the equation
y 00 = p(x)y 0 + q(x)y
converges wherever the power series for p(x) and q(x) both converge.
7. Consider the equation l(y) = a(x)y 00 + b(x)y 0 + c(x)y = h(x). Suppose that
we already know a basis for Ker(l): f1 (x) and f2 (x).
R∞
Show that yp (x) = −∞ G(x, t)h(t)dt is a solution to l(yp ) = h(x), where
the Green’s function is given by
G(x, t) =
f2 (x)f1 (t) − f1 (x)f2 (t)
[H(x − t) − H(x0 − t)].
a(t)W (t)
W (x) is the Wronskian; H(x) is the Heaviside step function.
Take your choice: you may either derive the solution using variation of
parameters or prove it by differentation.
2

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