Math 334
Assignment 4
Due: 12 Noon on Thursday, October 12, 2006.
1. The Wronskian of three functions is defined in terms of a determinant as follows:
W[φ1, φ2, φ3](x) :=
φ1(x)φ2(x)φ3(x)
φ′
1(x)φ′
2(x)φ′
3(x)
φ′′
1(x)φ′′
2(x)φ′′
3(x)
.
(a) Find the Wronskian of the functions
φ1(x) = 1, φ2(x) = x, φ3(x) = x2.
(b) Find the Wronskian of the functions
φ1(x) = ex, φ2(x) = e−x, φ3(x) = cosh x.
2. Use variation of parameters to find the general solution of
(a) y′′ + 16y= sec 4x;
(b) x2y′′ +xy′+ 9y=−tan(3 ln x).
3. Use variation of parameters to show that
y(x) = c1cos x+c2sin x+Zx
0
f(s) sin(x−s)ds
is the general solution to the differential equation
y′′ +y=f(x).
4. Determine the motion for an undamped system at resonance governed by
d2x
dt2+ 9x= 2 cos 3t, x(0) = 1,dx
dt (0) = 0,
5. Consider the following homogeneous boundary value problem on the interval (0,1) with homogeneous
boundary conditions:
y′′ +ω2y= 0, y(0) = 0, y(1) = 0.
The difference between initial value problems (IVPs) and boundary value problems (BVPs) is that the
auxiliary conditions for IVPs are applied at one point only, whereas the auxiliary conditions for BVPs
are applied at more than one point. While we have a theorem that guarantees that there is one and
only one solution for an IVP, the situation for BVPs is quite different. The trivial solution y≡0 is
always a solution to a homogeneous BVP, but there may be other solutions. In fact, there may be
infinitely many solutions.
Determine all the values of ωfor which the above BVP has at least one nontrivial solution.
6. Suppose φ1(x) and φ2(x) are linearly independent solutions of y′′ +P(x)y′+Q(x)y= 0. Suppose fur-
ther that φ1(x) has at least two zeros. Show that φ2(x) has one and only one zero between consecutive
zeros of φ1(x).