Gilbert Strang; Edwin “Jed” Herman

Review Exercises

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

262.

The differential equation $y' = 3 x^{2} y - cos (x) y^{″}$ is linear.

263.

The differential equation $y' = x - y$ is separable.

264.

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

265.

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.

266.

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

267.

$y' = 2^{x} + {cos}^{−1} x$

268.

$y' = y (x^{2} + 1)$

269.

$y' = e^{− y} sin x$

270.

$y' = 3 x - 2 y$

271.

$y' = y ln y$

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

272.

$y' = 8 x - ln x - 3 x^{4}, y (1) = 5$

273.

$y' = 3 x - cos x + 2, y (0) = 4$

274.

$x y' = y (x - 2), y (1) = 3$

275.

$y' = 3 y^{2} (x + cos x), y (0) = −2$

276.

$(x - 1) y' = y - 2, y (0) = 0$

277.

$y' = 3 y - x + 6 x^{2}, y (0) = −1$

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.

278.

$y' = 2 y - y^{2}$

279.

$y' = \frac{1}{x} + ln x - y,$ for $x > 0$

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?

280.

$y' = −4 y x, y (0) = 1$

281.

$y' = 3^{x} - 2 y, y (0) = 0$

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

282.

A car drives along a freeway, accelerating according to $a = 5 sin (π t),$ where $t$ represents time in minutes. Find the velocity at any time $t,$ assuming the car starts with an initial speed of $60$ mph.

283.

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 $g = 9.8 {m/s}^{2} .$

284.

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?

285.

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?

286.

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?

287.

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 $t .$

288.

You boil water to make tea. When you pour the water into your teapot, the temperature is $100 °C.$ After $5$ minutes in your $15 °C$ room, the temperature of the tea is $85 °C.$ 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 $(72 °C)?$

289.

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 $2 %$ 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 $1,998,257 ?$

290.

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