MTH 142 Practice Problems for Exam 3

This is a selection of sample problems from sections 10.1 through 10.7

1. Consider the differential equation $y^\prime = 2 - x - y$, where $a$ is a positive constant. One (and only one) of the following slope field plots is definitely not associated with this differential equation. Which plot does not correspond to the differential equation? Explain why not.
   
   
   
   
   
   
   

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2. Suppose that $y(x)$ is a solution of the initial value problem $y(0.2) = 1$, $y^\prime = x^2 + y$. Use Euler's method with $\Delta x = 0.1$ to estimate $y(0.5)$.
3. Make a rough sketch of the slope field for the differential equation $\frac{dy}{dx} = 2.5 y - 7.5$. Find an equilibrium solution and determine from the plot if it is stable or unstable. Explain why.

4. A yam has been heated to 170 degrees Fahrenheit and is placed in a room whose temperature is maintained at $70$ degrees F. In half an hour the yam cools to 100 degrees F. Let $Y(t)$ be the temperature of the yam at time $t$ hours after it was placed in the room.

   a) Assume the Newton's law of cooling applies, write a differential equation satisfied by $Y(t)$.

   b) Solve the differential equation to find a formula for $Y(t)$.

5. A drug is given intravenously at a rate of 5 mg/hr. The drug is excreted frmo the patient's body at a rate proportional to the amount present, with constant of proportionality equal to 0.2. No drug is present at time 0.

   a) Write a differential equation satisfied by $Q(t)$.

   b) If treatment is continued for a long, long time, how much of the drug will be present in the body?

6. Solve the initial value problems:

   a) $y \frac{dy}{dx} - \frac{x}{y} = 0$, $y(1) = 2$.

   b) $\frac{dr}{dx} + x e^r= x e^r$, $r(1) = 0$.

7. The growth of certain population is described by the differential equation $\frac{dP}{dt} = 0.001 (800 P - P^2)$, where $P(t)$ is the population at time $t$ days. The initial population is 25 mill.

   a) What is the population at the time when the population is growing most rapidly?

   b) Will the population ever exceed 900 mill.? Explain.

8. The following is data corresponding to certain population.

   t (days)	0	1	2	3	4	5	6
   P (mill)	3.975	7.2	12.825	23.55	37.65	57.0	79.27
   $\frac{dP}{dt}$
   $\frac{1}{P} \frac{dP}{dt}$

   
   

   a) Use the data to calculate approximately the blanks in the table.

   b) Use (a) to plot $\frac{1}{P} \frac{dP}{dt}$ versus $P$,

   c) fit a line as best you can through the points, and then obtain a differential equation satisfied by $P$.