MA2051 - Ordinary Differential Equations
Solutions to Sample Exam #1 - A96

A)

A kernels per minute are added.

B)

Kernels are added over minutes.

C)

Let k be the fraction of the kernels which pop each minute. Then k n(t) kernels per minute are popping out of the kettle.

D)

kernels pop out of the kettle in minutes.

E)

Conservation of popcorn: . Divide by and take the limit to obtain the differential equation n' = -kn + A. Add the initial condition to complete the model.

Solve to find years to double the world's population. (Just three generations!)

A)

To find a particular solution, use undetermined coefficients. Begin with . Substitute in y' = ky - 2t and collect coefficients of powers of t to find A - kB = 0, kA - 2 = 0. Hence, A = 2/k, , and .

To find a homogeneous solution, solve y' = ky by separation of variables or characteristic equations to find . Hence, a general solution is .

B)

Solve to find C = 1. The solution of the initial value problem is .

C)

Given the particular solution Z(t), use the homogeneous solution from above to write . To solve the initial value problem, solve to find C = 10. The solution of the initial value problem is .

A)

Solve to find the single steady state . To assess stability, examine a direction field diagram, graph solutions in the vicinity of , or use the following stability analysis.

B)

Consider a perturbation p(t) of the steady state. Let . Substitute in y' = 2y - M to find p' = 2p or . Since the perturbations grow, the steady state is unstable.