Math 334
Assignment 5
Due: 12 Noon on Thursday, October 19, 2006.
1. Find the solution to the following initial value problems.
(a) y′′′ −y′′ −4y′+ 4y= 0; y(0) = −4, y′(0) = −1, y′′(0) = −19.
(b) y′′′ −4y′′ + 7y′−6y= 0; y(0) = 1, y′(0) = 0, y′′(0) = 0.
2. Find the general solution to the following system of two second order differential equations:
d2x
dt2−x+y= 0, x +d2y
dt2−y=e3t.
3. Consider the fourth order differential equation:
y(iv)(x)−k2y′′(x) = g(x).(1)
(a) Find the general solution to the homogeneous equation.
(b) Show that a particular solution to Eq. (1) can be written in the form
φp(x) = 1
k2Zxg(x)dx −
x
k2Zg(x)dx +ekx
2k3Zg(x)e−kx dx −
e−kx
2k3Zg(x)ekx dx.
(c) Show that the general solution in part (b) can be rewritten in the form
φp(x) = Zx
0
g(s)G(x−s)ds,
where
G(ξ) = 1
k3(sinh kξ −kξ ).
(d) Determine the simplest form of the general solution to Eq. (1) in the case g(x)≡1.