6) (8 pts, 4 pts each) State the order of each ODE, thenclassify each of them as
linear/nonlinear, homogeneous/inhomogeneous, andautonomous/nonautonomous.
A) Unforced Pendulum: ??? + ? ?? + ?^2sin ? = 0

B) Simple RLC Circuit with a 9V Battery: Lq?? + Rq? +(1/c)q = 9

7) (8 pts) Find all critical points for the given DE, draw a phaseline for the system,
then state the stability of each critical point.
Logistic Equation: y? = ry(1 ? y/K), where r < 0

8) (6 pts) A mass of 2 kg is attached to the end of a spring and isacted on by an
external, driving force of 8 sin(t) N. When in motion, it movesthrough a medium that
imparts a viscous force of 4 N when the speed of the mass is 0.1m/s. The spring
constant is given as 3 N/m, and this mass-spring system is set intomotion from its
equilibrium position with a downward initial velocity of 1 m/s.Formulate the IVP
describing the motion of the mass. DO NOT SOLVE THE IVP.

9) (8 pts, 4 pts each) Find the maximal interval of existence, I,for each IVP given.
A) (t^2 ? 9) y? ? 7t^3 =?t, y(?2) = 12

B) sin(t) y?? + ty? ? 18y = 1, y(4) = 9, y?(4) = ?13


10) (30 pts, 10 pts each) Solve for the general solution to each ofthe DEs given. Use an
appropriate method in each case.
A) Newton’s Law of Cooling: y? = ?k(y ? T)

B) (sin(y) ? y sin(t)) dt + (cos(t) + t cos(y) ? y) dy = 0

C) ty? ? 5y = t^6 *e^t

11) (30 pts, 10 pts each) Solve for the general solution to each ofthe DEs given, then
classify the stability and type of critical point that lies at theorigin for each case.
A) y?? + y? ? 132y = 0

B) y?? + 361y = 0

C) y?? + 6y? + 10y = 0

12) (10 pts) Solve for the general solution to the DE given.
y?? ? 9y = ?18t^2 + 6