Week 4 Worksheet
Problem 1.
Consider the differential equation
y ′′ − y = 2 − x2 .
(a) Verify that ex and e−x solve the associated homogeneous equation and that yp = x2 is a particular solution
of the nonhomogeneous equation.
(b) Write the general solution of the nonhomogeneous equation.
(c) Find the solution satisfying
y ′ (0) = 0.
y(0) = 3,
Problem 2.
Let
L[y] = y ′′ + y.
(a) Verify that
[︃
L[x] = x
and
]︃
1
L − cos 2x = cos 2x.
3
(b) Use linearity to construct a particular solution of
y ′′ + y = 3x − 2 cos 2x.
(c) Find the solution satisfying
5
y(0) = ,
3
y ′ (0) = 1.
Problem 3.
Use the method of judicious guessing to solve the initial-value problem
y ′′ − 3y ′ + 2y = 4x + e3x ,
Problem 4.
7
y(0) = ,
2
5
y ′ (0) = .
2
Use the method of judicious guessing to solve the initial-value problem
y ′′ + 4y = 8 cos 2x,
y(0) = 1,
y ′ (0) = 0.
Explain why the usual trial function A cos 2x + B sin 2x cannot be used without modification.