Question

1) State the main difference between an ODE and a PDE?

2) Name two of the three archetypal PDEs?

3) Write the equation used to compute the Wronskian for two differentiable
functions, y1 and y2.

4) What can you conclude about two differentiable functions, y1 and y2, if their
Wronskian is nonzero?
5) (2 pts) If two functions, y1 and y2, solve a 2nd order DE, what does the Principle of
Superposition guarantee?


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

11) (30 pts, 10 pts each) Solve for the general solution to each of the DEs given, then
classify the stability and type of critical point that lies at the origin 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

13) (15 pts) Solve the given IVP.
y′′ + 2y′ + 2y = 10 sin(2t), y(0) = 1, y′(0) = 0

Answer 1