MATH 251 EXAM I, SPRING 2003

Solutions

1. i. linear, order 2

   ii. nonlinear, order 1

   iii. linear, order 3

   iv. nonlinear, order 2

2. A

3. C

4. C

5. D

6. (a) equilibrium solutions: y = -2, 0, 2

   (b)

   y = -2, (asymptotically) stable

   y = 0, unstable

   y = 2, (asymptotically) stable

   (c) y(t) = 2

7. (a) Q(t) = -6e^-t/200 + 6

   (b) it will never happen (or, when t approaches infinity)

8. (b) x ln(y) + x²y + sin(2x) = C

9. (a) yes

   (b) yes: W(y₁,y₂) = -4t, which is not equal to zero when t is not zero

   (c) since y₁ and y₂ are two linearly independent solutions, the general solution is therefore y(t) = C₁t³ + C₂t ^-1

10. (a) y(t) = C₁e^-4t + C₂e^t

    (b) y(t) = C₁e^-4t + C₂e^t - 3cos(2t) - 4sin(2t)

11. y(t) = 3e^-3t + 8te^-3t