MATH 251
Fall 2003
Exam 1
October 14, 2003
ANSWERS:
1. First order, linear;
Second order, linear;
First order, nonlinear;
Third order, nonlinear.
2. C
3. A
4. D
5. y(t) = โ4 + px4
โ3x2+ 3x+ 3.
6. (a) โ
โy (2x+yexy ) = exy +xyexy =โ
โx (xexy + 1)
(b) x2+exy +y= 2
7. (a) Equilibrium solutions are y=โ3,y= 0, and y= 2.
(b) y=โ3is unstable, y= 0 is (asymptotically) stable, y= 2 is unstable.
(c) lim
tโโ
y(t) = 0
(d) lim
tโโ
y(t) = โ3
8. (a) Q(t) = 25000 โ20000eโ1
100 t
(b) The limiting concentration is lim
tโโ
Q(t)
500 =25000
500 = 50 ( g
m3), which is the same as the concentration
of the inflow.
9. y(t) = 2e3tโte3t
10. y(t) = C1t+C2t4