## Review for the fourth part of the course

This review focuses on the material taught after the third midterm. It is basically all the applications to modeling, plus the techniques associated with systems of differential equations, and exact equations.

1. A body with mass $m=0.5$ kilogram is attached to the end of a spring that is stretched 2 meters by a force of 100 newtons. It is set in motion with initial position $x_0=1$ meters and initial velocity $v_0=-5 m/s.$ Find the position of the body as well as the amplitude, frequency, period of oscillation and time lag of its motion.
2. For the circuit below, suppose that $L=5 H, R=25 \Omega,$ and the source $E$ of emf is a battery supplying 100 V to the circuit. Suppose also that the switch has been in position 1 for a long time, so that a steady current of 4 A is flowing in the circuit. At time $t=0$ the switch is thrown to position 2, so that $I(0)=4$ and $E=0$ for $t \geq 0.$ Find $I(t).$

In the same circuit, now with the switch in position 1, suppose that $L=2, R=40, E(t)=100e^{-10t}$ and $I(0)=0$. Find the maximum current in the circuit for $t \geq 0.$

3. Consider two brine tanks connected as shown below. Tank 1 contains $x(t)$ pounds of salt in 100 gal of brine and tank 2 contains $y(t)$ pounds of salt in 200 gal of brine. The brine in each tank is kept uniform by stirring, and brine is pumped from each tank to the other at the rates indicated. In addition, fresh water flows into tank 1 at 20 gal/min, and the brine in tank 2 flows out at 20 gal/min (so the total volume of brine in the two tanks remain constant). The salt concentrations in the two tanks are $x/100$ pounds per gallon and $y/200$ pounds per gallon, respectively. Find a system of differential equations that model the rates of change of the amount of salt in the two tanks, and solve it.
4. Find a general solution to the system
$\begin{array}{rl} x'-4x+3y &=0 \\ -6x+y'+7y &=0\end{array}$
5. Use Euler’s method for systems to approximate a solution to the initial value problem
$\begin{cases} x' &=3x-2y \\ y' &=5x-4y\end{cases}$

with initial condition $x(0)=y(0)=1$ and step size $h=1.$

6. Interpret the following system as describing the interaction of two species with populations $x$ and $y.$ Draw a slope field and describe how solutions seem to behave. Find the critical points, and determine their stability. Sketch trajectories in the neighborhood of each critical point.
$\begin{array}{rlrl} (1) & x'=x(1.5-x-0.5y) & (2) & x'=x(1.5-x-0.5y) \\ & y'=y(2-y-0.75x) & & y'=y(2-0.5y-1.5x)\end{array}$
7. Verify that the equation below is exact, and solve it.
$\displaystyle{ \frac{2x^{5/2} - 3y^{5/3}}{2x^{5/2}y^{2/3}}\, dx + \frac{3y^{5/2}-2x^{5/2}}{3x^{3/2}y^{5/3}}\, dy = 0}$
8. A spherical tank of radius 4 ft is full of gasoline when a circular bottom hole with radius 1 in. is opened. How long will be required for all the gasoline to drain from the tank?