DISCLAIMER: The following practice exam is intended to give you a very general idea about the kinds of questions that may appear on the exam. It should not be regarded as a comprehensive study guide for the exam. In particular, it is possible that topics not covered on this practice exam will appear on the exam and that topics covered on this practice exam will not appear on the exam. You are responsible for all the topics covered in class.

1. Suppose that a given tank has valves that allow fluid to flow in at a rate proportional to the volume of fluid in the tank and out at a rate proportional to the square of the volume of fluid in the tank.
   1. Set up a DE that models this situation (assume $a$ is the constant of proportionality for the rate of flow into the tank and $b$ is the constant of proportionality for the rate of flow out of the tank).

   2. Draw a slope field for the equation you found above and find the equilibrium solution(s) if there are any. Describe (in plain English) what happens to the volume of fluid in the tank as $t \rightarrow \infty$. Be as specific as you can.

   3. Assuming there are initially $V_0$ gallons of fluid in the tank, solve your DE.

   4. Now suppose an extra valve is opened which allows fluid to enter the tank at a constant rate of $C$ gallons per unit time. Set up a new DE to model the situation, draw a slope field, and find the equilibrium solution(s) if there are any.

2. Solve the following differential equations. If no initial conditions are specified, find the general solution.
   1. $y' = -e^{x+y}$

   2. $xy' + (x+1)y = e^{-x}$,     $y(\ln 2) = 0$

   3. $y''' + 3y'' + 3y' + y = e^{2x}$

   4. $y'' + y' + y = 0$,     $y(0)=0$,     $y'(0)=1$

3. Find a recursion formula for $${\int (\ln x)^n dx}$$, $n > 1$.

4. 1. Compute the Taylor series for $f(x) = \frac{1}{1-x}$ about $x=0$ and find its radius of convergence.

   2. Compute the Taylor series for $f(x) = \frac{1}{1+x^2}$ about $x=0$ and find its radius of convergence.

   3. Compute the Taylor series for $f(x) = \arctan x$ about $x=0$ and find its radius of convergence.

5. 1. Explain why the series $1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$ must converge.

   2. It turns out that the above series converges to $\frac{\pi}{4}$ and thus $\pi = 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \cdots$. Explain why this is not a very useful formula for estimating the numerical value of $\pi$.

6. Suppose $L$ is a linear operator on the space of all differentiable real-valued functions.
   1. We can define a linear operator $e^L$. How? (HINT: how would Taylor answer this question?)

   2. Let $y$ be the function given by $y(x)=3x^3+x$ and let $D$ denote the differential operator. What is $e^D(y)$?