Download MATH 308 Homework 6

Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2010
MATH 308 Homework 6
6.1. Determine all critical points of the given system of equations. Find the corresponding
linear system near each critical point. What conclusion can you then draw about the
nonlinear system?
dx/dt = x − y,
dy/dt = x − 2y + xy − 2.
6.2. Solve the given initial value problem. Describe behavior of the solution for increasing
t.
9y 00 + 12y 0 + 4y = 0,
y(0) = 2, y 0 (0) = −1.
6.3. Solve the given initial value problem. Describe behavior of the solution for increasing
t.
y 00 + 4y 0 + 3y = 0,
y(0) = 2, y 0 (0) = −1.
6.4. Use the method of reduction of order to find a second solution y2 of the differential
equation such that {y1 , y2 } is a fundamental set of solutions on the given interval.
t2 y 00 + 3ty 0 + y = 0,
t > 0;
6.5. Find the solution of the given initial value problems.
(a) y 00 + 4y = 0,
y(0) = 0,
(b) y 00 + 4y 0 + 5y = 0,
y 0 (0) = 1;
y(0) = 1,
y 0 (0) = 0.
6.6. Find the general solution of the equation for x > 0.
x2 y 00 − 2y = 0.
y1 = t−1 .