4.R: Chapter 4 Review Exercises

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

1) The differential equation \(y'=3x^2y−\cos(x)y''\) is linear.

2) The differential equation \(y'=x−y\) is separable.

Answer

\(F\)

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

3) \(y′=x^2+3e^x−2x\)

4) \(y'=2^x+\cos^{−1}x\)

Answer

\(y(x)=\frac{2^x}{ln(2)}+xcos^{−1}x−\sqrt{1−x^2}+C\)

5) \(y'=y(x^2+1)\)

6) \(y'=e^{−y}\sin x\)

Answer

\(y(x)=\ln(C−\cos x)\)

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

7) \(y'=8x−\ln x−3x^4, \quad y(1)=5\)

8) \(y'=3x−\cos x+2, \quad y(0)=4\)

Answer

\(y(x)=4+\frac{3}{2}x^2+2x−\sin x\)

9) \(xy'=y(x−2), \quad y(1)=3\)

10) \(y'=3y^2(x+\cos x), \quad y(0)=−2\)

Answer

\(y(x)=−\dfrac{2}{1+3(x^2+2\sin x)}\)

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?

11) \(y'=−4yx, \quad y(0)=1\)

12) \(y'=3^x−2y, \quad y(0)=0\)

Answer

Euler: \(0.6939\),
Exact solution: \(y(x)=\dfrac{3^x−e^{−2x}}{2+\ln(3)}\)

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

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

14) 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 \(g=9.8\,\text{m/s}^2\).

Answer

\(\frac{40}{49}\) second

15) 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?

16) 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 \(t\).

Answer

\(T(t)=200\left(1−e^{−t/1000}\right)\)

17) 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 (\(72°C\))?

18) 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 \(1,998,257\)?

Answer

\(P(t)=\dfrac{1600000e^{0.02t}}{9840+160e^{0.02t}}\)

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

19) \(y'=2y−y^2\)