TEXAS A&M UNIVERSITY
DEPARTMENT OF MATHEMATICS
MATH 308-502
Final Exam, 9 Dec 2013, 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.
1.
Solve the initial value problem
xy 0 = 3y + x3 ,
y(1) = 10.
2.
Find the general solution of
(3x2 + 2y 2 )dx + (4xy + 6y 2 )dy = 0
3.
Find the general solution of
x00 − 2x0 + 5x = sin 2t.
4.
Find the solution of the initial value problem
!
1 1
0
x =
x,
4 −2
x(0) =
!
3
.
−2
Identify the type of the critical point at (0, 0) and draw a phase portrait (orient the eigenvectors properly).
5.
Find the general solution and sketch the phase portrait of the system, identify the type of
the critical point at (0, 0)
!
−1
−4
x0 =
x.
1 −1
6.
Find the general solution of the non-homogeneous system

x0 = x + x + 5e2t ,
1
2
1
0
x = 4x1 − 2x2 .
2
(Hint: the homogeneous part is solved in a previous question).
7.
For the nonlinear system

x0 = 1 − 2xy,
y 0 = x − y 3 /2,
(2 points) find all critical points, (6 points) find their type and stability, (2 points) sketch
the phase portrait.
8.
Bonus question +2 points (no partial credit): Solve question 3 by a different method.
Points:
/40