### Review Exercises

*True or False?* Justify your answer with a proof or a counterexample.

The differential equation $y\prime =3{x}^{2}y-\text{cos}(x)y\text{\u2033}$ is linear.

You can explicitly solve all first-order differential equations by separation or by the method of integrating factors.

You can determine the behavior of all first-order differential equations using directional fields or Euler’s method.

For the following problems, find the general solution to the differential equations.

${y}^{\prime}={x}^{2}+3{e}^{x}-2x$

$y\prime =y\left({x}^{2}+1\right)$

$y\prime =3x-2y$

For the following problems, find the solution to the initial value problem.

$y\prime =8x-\text{ln}\phantom{\rule{0.1em}{0ex}}x-3{x}^{4},y(1)=5$

$xy\prime =y\left(x-2\right),y(1)=3$

$(x-1)y\prime =y-2,y(0)=0$

For the following problems, draw the directional field associated with the differential equation, then solve the differential equation. Draw a sample solution on the directional field.

$y\prime =2y-{y}^{2}$

For the following problems, use Euler’s Method with $n=5$ steps over the interval $t=[0,1].$ Then solve the initial-value problem exactly. How close is your Euler’s Method estimate?

$y\prime =\mathrm{-4}yx,y(0)=1$

For the following problems, set up and solve the differential equations.

A car drives along a freeway, accelerating according to $a=5\phantom{\rule{0.1em}{0ex}}\text{sin}(\pi t),$ where $t$ represents time in minutes. Find the velocity at any time $t,$ assuming the car starts with an initial speed of $60$ mph.

You throw a ball of mass $2$ kilograms into the air with an upward velocity of $8$ m/s. Find exactly the time the ball will remain in the air, assuming that gravity is given by $g=9.8{\phantom{\rule{0.2em}{0ex}}\text{m/s}}^{2}.$

You drop a ball with a mass of $5$ kilograms out an airplane window at a height of $5000$ m. How long does it take for the ball to reach the ground?

You drop the same ball of mass $5$ kilograms out of the same airplane window at the same height, except this time you assume a drag force proportional to the ball’s velocity, using a proportionality constant of $3$ and the ball reaches terminal velocity. Solve for the distance fallen as a function of time. How long does it take the ball to reach the ground?

A drug is administered to a patient every $24$ hours and is cleared at a rate proportional to the amount of drug left in the body, with proportionality constant $0.2.$ If the patient needs a baseline level of $5$ mg to be in the bloodstream at all times, how large should the dose be?

A $1000$-liter tank contains pure water and a solution of $0.2$ kg salt/L is pumped into the tank at a rate of $1$ L/min and is drained at the same rate. Solve for total amount of salt in the tank at time $t.$

You boil water to make tea. When you pour the water into your teapot, the temperature is $100\text{\xb0C.}$ After $5$ minutes in your $15\text{\xb0C}$ room, the temperature of the tea is $85\text{\xb0C.}$ Solve the equation to determine the temperatures of the tea at time $t.$ How long must you wait until the tea is at a drinkable temperature $(72\text{\xb0C})?$

The human population (in thousands) of Nevada in $1950$ was roughly $160.$ If the carrying capacity is estimated at $10$ million individuals, and assuming a growth rate of $2\text{\%}$ per year, develop a logistic growth model and solve for the population in Nevada at any time (use $1950$ as time = 0). What population does your model predict for $2000?$ How close is your prediction to the true value of $\mathrm{1,998,257}?$

Repeat the previous problem but use Gompertz growth model. Which is more accurate?