MATH 441 SECTION X13 Review Problems for the Second Exam
Coverage: Sections 2.8, 3.1-3.7, 4.1-4.2.
Problem 1 Solve the differential equation
t2y
′′
−4ty
′
−6y= 0.
Ans: y=c1t6+c2t−1.
Problem 2
Use the method of the reduction of order to find a second solution of the differ-
ential equation for x > 1 if y1(x) = exis a solution.
(x−1)y
′′
−xy
′+y= 0.
Ans: y2(x) = x.
Problem 3
Find the solution of the initial value problem
y
′′ + 4y= 3 sin(2t)y(0) = 2, y
′(0) = −1.
Ans: y= 2 co s(2t)−
1
8sin(2t)−
3
4tcos(2t).
Problem 4 Show that y1(x) = x2and y2(x) = x2ln xsatisfy the homogeneous
equation and then find a particular solution of the nonhomogeneous equation
for x > 0,
x2y
′′
−3xy
′+ 4y=x2ln x.
Ans: y=1
6x2ln3(x).
Problem 5 Find the solution of the initial value problem,
y
′′′ +y
′= 0, y(0) = 0, y
′(0) = 1, y
′′ (0) = 2.
Ans: y= 2 −2 cos(t) + sin(t).
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