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 + C2et

    (b) y(t) = C1e-4t + C2et - 3cos(2t) - 4sin(2t)

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