**MATH
251 EXAM I, SPRING 2003**

Solutions

i. linear, order 2

ii. nonlinear, order 1

iii. linear, order 3

iv. nonlinear, order 2

A

C

C

D

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

(b)

y = -2, (asymptotically) stable

y = 0, unstable

y = 2, (asymptotically) stable

(c) y(t) = 2

(a) Q(t) = -6e

^{-t/200}+ 6(b) it will never happen (or, when t approaches infinity)

(b) x ln(y) + x

^{2}y + sin(2x) = C(a) yes

(b) yes: W(y

_{1},y_{2}) = -4t, which is not equal to zero when t is not zero(c)

*since y*, the general solution is therefore y(t) = C_{1}and y_{2}are two linearly independent solutions_{1}t^{3}+ C_{2}t^{-1}(a) y(t) = C

_{1}e^{-4t}+ C_{2}e^{t}(b) y(t) = C

_{1}e^{-4t}+ C_{2}e^{t}- 3cos(2t) - 4sin(2t)y(t) = 3e

^{-3t}+ 8te^{-3t}