MATH 251 Midterm: MATH 251 PSU s251Exam 1(fa04)
MATH 251
Fall 2004
Exam I
October 12, 2004
ANSWERS:
1. (a) first order, non-linear; (b) second order, non-linear; (c) second order, linear; (d) third order, linear.
2. C
3. C
4. D
5. A
6. C
7. y(t) = −te−t−e−t+ 2e−1
t4.
8. (a) The initial value problem describing the process is: Q′= 3e−t−3
100 Q, Q(0) = 100.
The solution is Q(t) = 10000
97 e
−3
100 t−300
97 e−t.
(b) The limiting concentration is lim
t→∞
Q(t)
100 =1
100 lim
t→∞
Q(t) = 0.
9. (a) ∂(4xy −3)
∂x = 4y=∂(2y2+ 2x)
∂y ⇒it’s an exact equation.
(b) 2xy2−3y+x2=C.
10. (a) y(t) = −1
2e−tsin 2t.
(b) y(t) = t2
4+ 2 ln t−1−4 ln 2.
11. α=−1
4.
12. (a) y2=t4.
(b) W(t, t4) = 3t4>0through out the interval (0,∞).
