## Exam 1

Math 256 Exam 1 part 1 Summer 2012

Please attach ALL of your work if full or partial credit is
unless otherwise noted. Enjoy!

1. Population Growth
1. Easy
The population of the United State of America was about 3.93 million in 1790, 17.07 million in 1840, and 62.98 million in 1890. Estimating the carrying capacity of the US at $K=989.5$ million and the intrinsic growth rate at $r=3.0463\%$, find a function for the US population over time based on the logistic differential equation:
$\frac{dy}{dt}=ry\left(1-\frac{y}{K}\right)$
2. Medium
Consider a local population of frogs with an intrinsic growth rate, $r=0.5$, and a carrying capacity, $K=3$ (in hundreds).
$\frac{dy}{dt}=ry\left(1-\frac{y}{K}\right)$

1. Create a slope field for $x\in[0,10]$ and $y\in[-1,5]$
2. Identify equilibrium solutions, i.e. critical points:
If $y'=f(y)$, then any $y$ for which $f(y)=0$ is a critical point.
3. Sketch 5 integral curves through the slope field.
4. Find a general solution for the differential equation.
3. Hard
$\displaystyle \frac{dP}{dt}=rP\left(1-\frac{P}{C}\right)$ , $P(0)=76.2$ , $t=0$ refers to $1900$, $r$ is the intrinsic growth rate, and C is the predicted Carrying Capacity.
US Population (Census Bureau)

 date $t$ US Population in millions 1900 0 76.2 1950 50 151.3 2000 100 282.4

Based on the differential equation and boundary values, predict the US population in 2010 and 2050.

2.  Suppose that a cup of soup is cooled from $90^{\circ}$C to $60^{\circ}$C after 10 minutes in a room whose temperature was $20^{\circ}$C. Newton's law of cooling states that the rate of change of the temperature of an object is proportional to the difference between the environmental temperature and the objects temperature.
1. Write an initial value problem for this situation.
2. Solve the initial value problem.
3. How much time will it take the cup of soup to cool from $90^{\circ}$C to $35^{\circ}$C?
3. A tank initially contains 5 gallons of sugar water in which 12 lb of sugar is dissolved. Pure water is flowing in at a rate of 1 gal/hr. The mixture is kept uniform by stirring and flows out at a rate of 2 gal/hr.
1. What is the volume of the tank at time $t$ hours?
2. At what rate (pounds per hour) does sugar enter the take?
3. At what rate (pounds per hour) does sugar leave the take?
4. Write down and solve an initial value problem describing the mixing process.
5. Find the amount of sugar in the tank 4 hours after the process begins.
4. A ball with a mass of 15 kg is thrown upward with an initial velocity of 20 m/sec from the roof of a 30m high building. Assume that air resistance is $\frac{|v|}{30}$.
1. Find the maximum height above the ground that the ball reaches.
2. Find the time it take for the ball to hit the ground.
3. Plot the graphs of velocity and position versus time.
5. An RL circuit, $\displaystyle L\frac{dI}{dt}+RI=E(t)$, with a 1-$\Omega$, $R$, resistor and a $0.01$-H inductor, $L$, is driven by a voltage of $E(t)=\sin(100t)$ V. If the initial inductor current is zero, determine the subsequent resistor and inductor voltages ($L\frac{dI}{dt}$ and $I(t)$) and the current ($I(t)$).
6. Miscellaneous Problems:
1. $\displaystyle (x+y)dx-(x-y)dy=0$
2. $\displaystyle \frac{dy}{dx}=\frac{2x+y}{3+3y^2-x}$, $y(0)=0$
3. $\displaystyle \frac{dy}{dx}=-\frac{2xy+1}{x^2+2y}$
4. $\displaystyle x\frac{dy}{dx}+2y=\frac{\sin(x)}{x}$, $y(1)=0$
5. $\displaystyle \frac{dy}{dx}+y=\frac{1}{1+e^x}$
6. $\displaystyle (x+y)dx+(x+2y)dy=0$, $y(2)=3$
7. $\displaystyle \frac{dy}{dx}=\frac{x^3-2y}{x}$
8. $\displaystyle \frac{dy}{dx}=\frac{x^2+y^2}{x^2}$
9. $\displaystyle y'=e^{x+y}$
10. $\displaystyle 2\sin(y)\cos(x)dx+\cos(y)\sin(x)dy=0$
11. $\displaystyle xy'+y-y^2e^{2x}=0$
12. $\displaystyle xdy-ydx=2x^2y^2dy$, $y(1)=-2$
13. $\displaystyle \frac{dy}{dx}=\sqrt{y^2-1}$