Question

6) (8 pts, 4 pts each) State the order of each ODE, then classify each of them as
linear/nonlinear, homogeneous/inhomogeneous, and autonomous/nonautonomous.
A) Unforced Pendulum: θ′′ + γ θ′ + ω^2sin θ = 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 phase line 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 is acted on by an
external, driving force of 8 sin(t) N. When in motion, it moves through a medium that
imparts a viscous force of 4 N when the speed of the mass is 0.1 m/s. The spring
constant is given as 3 N/m, and this mass-spring system is set into motion 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 of the 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

Answer 1