MATH 251
Summer 2006
Exam I
June 29, 2006
ANSWERS:
1. (a) Third order, linear; (b) First order, nonlinear; (c) Second order, nonlinear.
2. C
3. C
4. B
5. A
6. (a) y(t) = t2sin t+2
π2t2; (b) (0,∞).
7. (a) y= 6,5,−5; (b) y=−5is (asymptotically) stable, y= 5 is unstable, y= 6 is semistable;
(c) lim
t→∞
y(t) = −5; (d) y0= 5.
8. (a) ∂M
∂y =−2x+ex+y=∂ N
∂x ; (b) 2x3−x2y+ex+y= 4.
9. y(t) = 3e−2tcos 3t−e−2tsin 3t
10. y(t) = C1t+C2tln t
11. (a) Q0= 100 −1
200Q, Q(0) = 0; (b) Q(t) = 20000 −20000e
−t
200 ;
(c) t= 200ln 2; (d) lim
t→∞
Q(t) = 20000 g.