Math 142 Exam 2 Summer 2018
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Math 142
Exam 2
Summer 2018
1. (a) (10 points) The Logistic equation 
= y(2 − .1y), y(0) = 5 has solution
Prove that another way to write this solution is y = 10 + 10 tanh(t − t0 ) where t0 = 
ln(3) ≈ .55.
(b) (5 points) Use the above to give a detailed sketch of y(t). Label any interesting points on the axes.
2. (10 points) Suppose
Show that y(t) = y0eR(t)t, where R(t) is the average value of f on [0, t].
3. A population of mice 
has 3 components, of ages 0-1 years, 1-2 years, and 2-3 years. The birth and death rates of each population in percentage of mice/year is:
(a) (4 points) Find a matrix A such that 
(b) (4 points) Show that A satisfies the hypotheses of the Perron-Frobenious Theorem.
(c) (4 points) The eigenvalues and eigenvectors of your matrix A should be
Find the general form of the solution for Nk if the initial population is 
(Don’t just write the answer. Show how you get it.)
(d) (4 points) What is the percentage of mice of aged 1-2 years among the total pop- ulation as k → ∞?
(e) (4 points) Explain how the answer from part c) gives you the answer in part d).
4.
Nk+1 − Nk = Nk − 1 (2Nk − 1 − 6) .
(a) (4 points) Find the critical points.
(b) (8 points) Find the linearized equation around the positive critical point.
(c) (8 points) Solve xk+1 − xk − 6xk − 1 = 0, x0 = 1, x1 = 0.
(d) (4 points) Is your critical point from b) stable? Explain.
5. Consider the Delay Differential equation
(a) (8 points) Show that there is a solution of the form y = eλt for λ real if and only if 
(b) (8 points) For 
, there are solutions eλt for λ complex. Find the value of td
such that λ is purely imaginary. What are the real solutions y(t) in this case?
For the nonlinear system
2023-07-25