TEXAS A&M UNIVERSITY
DEPARTMENT OF MATHEMATICS
MATH 308-502
Final Exam, 15 Dec 2014, version A
On my honor, as an Aggie, I have neither given nor received unauthorized aid on this work.
Name (print):
In all questions, no analytical work — no points.
Each question is 5 points unless noted otherwise.
1.
Solve the initial value problem
3
y 0 + y = 4 ln(x),
x
3
y(1) = .
4
2.
Solve the initial value problem
dy
cos(x − y) + 2x
=
,
dx
cos(x − y) − 3y 2
y(π) = 0.
(Hint: This equation is either linear, separable or exact.)
3.
Find the general solution of
x00 − x0 − 6x = et .
4.
Find the general solution of
x00 + 4x = cos(2t).
5.
For the system
0
x =
!
2 1
x,
1 α
and the critical point at (0, 0),
1. let α = 0; find eigenvalues, eigenvectors, identify the type of the critical point and
draw the phase portrait.
2. Do the same for α = 2.
3. At which value of α between 0 and 2 does the portrait undergo qualitative change?
Bonus +2 points (no partial credit): Draw and explain the phase portrait at the
change point.
6.
(10 points) For the competing species system

x0 = x(5 − x − 2y),
y 0 = y(3 − y − x),
find all critical points, their type (find eigenvalues and eigenvectors) and sketch the phase
portrait. Can the species coexist in a stable manner?
Points:
/35