# TEXAS A&amp;M UNIVERSITY DEPARTMENT OF MATHEMATICS MATH 308-200 ```TEXAS A&amp;M UNIVERSITY
DEPARTMENT OF MATHEMATICS
MATH 308-200
Exam 1 version A, 29 Sep 2011
On my honor, as an Aggie, I have neither given nor received unauthorized aid on this work.
Name (print):
No detailed analytical work — no points.
1.
Find the general solution of
2
xy 0 − (2x2 + 1)y = −ex .
2.
Find the general solution to
1
3
y 0 = 0;
3x sin(2y) + 2x cos(2y) +
y
2
3.
For the equation
y 0 = y(1 − y),
1. Find stationary solutions (critical points) and draw the phase diagram.
2. Using the phase diagram, predict the large t limit of the solution y(t) that satisfies
the initial condition y(0) = 1/2.
3. Find explicit solution to the IVP and compare with your prediction.
4.
The small oscillations of a pendulum are described by the equation
θ00 + bθ0 + 16θ = 0,
where θ(t) is the angle and b is related to friction.
1. Find the general solution for b = 0, 4, 8, 10.
2. For which of the above values the solution tends to 0? Explain the physical meaning