CARNEGIE MELLON UNIVERSITY
DEPARTMENT OF MATHEMATICAL SCIENCES
21-122 Review Exam 2, Spring, 2006

• Your exam shall consist of problems similar to the homework and quiz problems.
• Monday, March 6th will be a review day. Please ask any questions you may have from the sections 8.1,9.1-9.7,17.1.
• The exam will be held on Wednesday March 8th. All students will take the exam in PH100 during their assigned lecture time.
• Notes and texts will not be permitted during the exam.
• Calculators or other electronic devices are not permitted.
• You will have 50 minutes for the exam, and must stop when told to stop.
• Please consult the course web page for exam policies - these will be enforced. What follows is a practice exam. The actual exam will consist of different questions. To fully prepare please practice many homework problems.

1. Find the arc length of $f(x)=\ln(\sec \theta)$, $0 \leq \theta \leq \pi/3$.

2. Find the general solution of $\frac{dy}{dx} + e^x y = e^x y^2$

3. Find the general solution of $\frac{d^2y}{dx^2}-5\frac{dy}{dx}=e^{4x}$

4. Determine the value $y_1$ for which the following boundary-value problem has solutions, and describe all these possible solutions. $\frac{d^2y}{dx^2}+2\frac{dy}{dx}+10y=0$, $y(0)=1$, $y(\pi)=y_1$

5. Determine the equilibrium solutions to $\frac{dy}{dx}=y/3-y^2/2+y^3/6$, and classify them as stable or unstable.

6. Suppose $10\%$ of a radioactive substance decays in one year. What percentage of the substance decays in two years?

7. Consider the logistic equation with harvesting: $\frac{dP}{dt}=0.2P(1-\frac{P}{200})-c$. Determine the values of $c$ for which the population will always die out.

8. Consider the predator-prey equations:
   
   $$\frac{dR}{dt}=0.5R-0.025RW, \ \frac{dW}{dt}=-.25W+.00125RW$$
   
   
   1. Find the non-trivial equilibrium solutions.
   2. Determine whether the rabbit and wolf poulations are increasing or decreasing if there are currently 100 rabbits and 10 wolves.
   3. Sketch the phase trajectory in the phase plane corresponding to an initial population of 100 rabbits and 10 wolves.