Calculus

Gilbert Strang,

Chapter 4

Introduction to Differential Equations - all with Video Answers

Educators

DE

MA

Section 5

First-order Linear Equations

Problem 208

Are the following differential equations linear? Explain your reasoning.
$$
\frac{d y}{d x}=x^{2} y+\sin x
$$

Prathan J.

Prathan J.

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Problem 209

Are the following differential equations linear? Explain your reasoning.
$$
\frac{d y}{d t}=t y
$$

Robert D.

Numerade Educator

Problem 210

Are the following differential equations linear? Explain your reasoning.
$$
\frac{d y}{d t}+y^{2}=x
$$

Prathan J.

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Problem 211

Are the following differential equations linear? Explain your reasoning.
$$
y^{\prime}=x^{3}+e^{x}
$$

Robert D.

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Problem 212

Are the following differential equations linear? Explain your reasoning.
$$
y^{\prime}=y+e^{y}
$$

Prathan J.

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Problem 213

Write the following first-order differential equations in standard form.
$$
y^{\prime}=x^{3} y+\sin x
$$

DE

David E.

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Problem 214

Write the following first-order differential equations in standard form.
$$
y^{\prime}+3 y-\ln x=0
$$

Prathan J.

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Problem 215

Write the following first-order differential equations in standard form.
$$
-x y^{\prime}=(3 x+2) y+x e^{x}
$$

Robert D.

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Problem 216

Write the following first-order differential equations in standard form.
$$
\frac{d y}{d t}=4 y+t y+\tan t
$$

Prathan J.

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Problem 217

Write the following first-order differential equations in standard form.
$\frac{d y}{d t}=y x(x+1)$

MA

Melissa A.

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Problem 218

What are the integrating factors for the following differential equations?
$$
y^{\prime}=x y+3
$$

Prathan J.

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Problem 219

What are the integrating factors for the following differential equations?
$$
y^{\prime}+e^{x} y=\sin x
$$

Robert D.

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Problem 220

What are the integrating factors for the following differential equations?
$$
y^{\prime}=x \ln (x) y+3 x
$$

Prathan J.

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Problem 221

What are the integrating factors for the following differential equations?
$$
\frac{d y}{d x}=\tanh (x) y+1
$$

Robert D.

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Problem 222

What are the integrating factors for the following differential equations?
$$
\frac{d y}{d t}+3 t y=e^{t} y
$$

Prathan J.

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Problem 223

Solve the following differential equations by using integrating factors.
$$
y^{\prime}=3 y+2
$$

Robert D.

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Problem 224

What are the integrating factors for the following differential equations?
$$
y^{\prime}=2 y-x^{2}
$$

Prathan J.

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Problem 225

What are the integrating factors for the following differential equations?
$$
x y^{\prime}=3 y-6 x^{2}
$$

Robert D.

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Problem 226

What are the integrating factors for the following differential equations?
$$
(x+2) y^{\prime}=3 x+y
$$

Prathan J.

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Problem 227

What are the integrating factors for the following differential equations?
$$
y^{\prime}=3 x+x y
$$

Robert D.

Numerade Educator

Problem 228

What are the integrating factors for the following differential equations?
$$
x y^{\prime}=x+y
$$

Prathan J.

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Problem 229

What are the integrating factors for the following differential equations?
$$
\sin (x) y^{\prime}=y+2 x
$$

Robert D.

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Problem 230

What are the integrating factors for the following differential equations?
$$
y^{\prime}=y+e^{x}
$$

Prathan J.

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Problem 231

What are the integrating factors for the following differential equations?
$$
x y^{\prime}=3 y+x^{2}
$$

Robert D.

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Problem 232

What are the integrating factors for the following differential equations?
$$
y^{\prime}+\ln x=\frac{y}{x}
$$

Prathan J.

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Problem 233

Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?
$$
(x+2) y^{\prime}=2 y-1
$$

Robert D.

Numerade Educator

Problem 234

Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?
$$
y^{\prime}=3 e^{t / 3}-2 y
$$

Robert D.

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Problem 235

Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?
$$
x y^{\prime}+\frac{y}{2}=\sin (3 t)
$$

Robert D.

Numerade Educator

Problem 236

Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?
$$
x y^{\prime}=2 \frac{\cos x}{x}-3 y
$$

Robert D.

Numerade Educator

Problem 237

Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?
$$
(x+1) y^{\prime}=3 y+x^{2}+2 x+1
$$

Robert D.

Numerade Educator

Problem 238

Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?
$$
\sin (x) y^{\prime}+\cos (x) y=2 x
$$

Robert D.

Numerade Educator

Problem 239

Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?
$$
\sqrt{x^{2}+1} y^{\prime}=y+2
$$

Robert D.

Numerade Educator

Problem 240

Solve the following differential equations. Use your calculator to draw a family of solutions. Are there certain initial conditions that change the behavior of the solution?
$$
x^{3} y^{\prime}+2 x^{2} y=x+1
$$

Robert D.

Numerade Educator

Problem 241

Solve the following initial-value problems by using integrating factors.
$$
y^{\prime}+y=x, y(0)=3
$$

Robert D.

Numerade Educator

Problem 242

Solve the following initial-value problems by using integrating factors.
$$
y^{\prime}=y+2 x^{2}, y(0)=0
$$

Prathan J.

Numerade Educator

Problem 243

Solve the following initial-value problems by using integrating factors.
$$
x y^{\prime}=y-3 x^{3}, y(1)=0
$$

Robert D.

Numerade Educator

Problem 244

Solve the following initial-value problems by using integrating factors.
$$
x^{2} y^{\prime}=x y-\ln x, y(1)=1
$$

Prathan J.

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Problem 245

Solve the following initial-value problems by using integrating factors.
$$
\left(1+x^{2}\right) y^{\prime}=y-1, y(0)=0
$$

Robert D.

Numerade Educator

Problem 246

Solve the following initial-value problems by using integrating factors.
$$
x y^{\prime}=y+2 x \ln x, y(1)=5
$$

Prathan J.

Numerade Educator

Problem 247

Solve the following initial-value problems by using integrating factors.
$$
(2+x) y^{\prime}=y+2+x, y(0)=0
$$

Robert D.

Numerade Educator

Problem 248

Solve the following initial-value problems by using integrating factors.
$$
y^{\prime}=x y+2 x e^{x}, y(0)=2
$$

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Problem 249

Solve the following initial-value problems by using integrating factors.
$$
\sqrt{x} y^{\prime}=y+2 x, y(0)=1
$$

Robert D.

Numerade Educator

Problem 250

Solve the following initial-value problems by using integrating factors.
$$
y^{\prime}=2 y+x e^{x}, y(0)=-1
$$

Prathan J.

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Problem 251

A falling object of mass $m$ can reach terminal velocity when the drag force is proportional to its velocity, with proportionality constant $k$ . Set up the differential equation and solve for the velocity given an initial velocity of $0 .$

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Problem 252

Using your expression from the preceding problem, what is the terminal velocity? (Hint: Examine the limiting behavior; does the velocity approach a value?)

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Problem 253

Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall 5000
meters if the mass is 100 kilograms, the acceleration due to gravity is 9.8 $\mathrm{m} / \mathrm{s}^{2}$ and the proportionality constant is 4$?$

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Problem 254

A more accurate way to describe terminal velocity is that the drag force is proportional to the square of velocity, with a proportionality constant $k$ . Set up the differential equation and solve for the velocity.

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Problem 255

Using your expression from the preceding problem. what is the terminal velocity? (Hint: Examine the limiting behavior: Does the velocity approach a value?)

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Problem 256

Using your equation for terminal velocity, solve for the distance fallen. How long does it take to fall 5000
meters if the mass is 100 kilograms, the acceleration due to gravity is 9.8 $\mathrm{m} / \mathrm{s}^{2}$ and the proportionality constant is 4$?$ Does it take more or less time than your initial estimate?

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Problem 257

For the following problems, determine how parameter $a$ affects the solution.
Solve the generic equation $y^{\prime}=a x+y .$ How does varying $a$ change the behavior?

Robert D.

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Problem 258

For the following problems, determine how parameter $a$ affects the solution.
Solve the generic equation $y^{\prime}=a x+y .$ How does varying $a$ change the behavior?

Robert D.

Numerade Educator

Problem 259

For the following problems, determine how parameter $a$ affects the solution.
Solve the generic equation $y^{\prime}=a x+x y .$ How does varying $a$ change the behavior?

Robert D.

Numerade Educator

Problem 260

For the following problems, determine how parameter $a$ affects the solution.
Solve the generic equation $y^{\prime}=x+a x y .$ How does varying $a$ change the behavior?

Robert D.

Numerade Educator

Problem 261

For the following problems, determine how parameter $a$ affects the solution.
Solve $y^{\prime}-y=e^{k t}$ with the initial condition $y(0)=0 .$ As $k$ approaches $1,$ what happens to your formula?

Robert D.

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Problem 262

True or False? Justify your answer with a proof or a counterexample.
The differential equation $y^{\prime}=3 x^{2} y-\cos (x) y^{\prime \prime}$ is linear.

Prathan J.

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Problem 263

True or False? Justify your answer with a proof or a counterexample.
The differential equation $y^{\prime}=x-y$ is separable.

Robert D.

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Problem 264

True or False? Justify your answer with a proof or a counterexample.
You can explicitly solve all first-order differential equations by separation or by the method of integrating
factors.

Prathan J.

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Problem 265

True or False? Justify your answer with a proof or a counterexample.
You can determine the behavior of all first-order differential equations using directional fields or Euler's
method.

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Problem 266

For the following problems, find the general solution to the differential equations.
$$
y^{\prime}=x^{2}+3 e^{x}-2 x
$$

Prathan J.

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Problem 267

For the following problems, find the general solution to the differential equations.
$$
y^{\prime}=2^{x}+\cos ^{-1} x
$$

Robert D.

Numerade Educator

Problem 268

For the following problems, find the general solution to the differential equations.
$$
y^{\prime}=y\left(x^{2}+1\right)
$$

Prathan J.

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Problem 269

For the following problems, find the general solution to the differential equations.
$$
y^{\prime}=e^{-y} \sin x
$$

Robert D.

Numerade Educator

Problem 270

For the following problems, find the general solution to the differential equations.
$$
y^{\prime}=3 x-2 y
$$

Prathan J.

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Problem 271

For the following problems, find the general solution to the differential equations.
$$
y^{\prime}=y \ln y
$$

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Problem 272

For the following problems, find the solution to the initial value problem.
$$
y^{\prime}=8 x-\ln x-3 x^{4}, y(1)=5
$$

Prathan J.

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Problem 273

For the following problems, find the solution to the initial value problem.
$$
y^{\prime}=3 x-\cos x+2, y(0)=4
$$

Robert D.

Numerade Educator

Problem 274

For the following problems, find the solution to the initial value problem.
$$
x y^{\prime}=y(x-2), y(1)=3
$$

Prathan J.

Numerade Educator

Problem 275

For the following problems, find the solution to the initial value problem.
$$
y^{\prime}=3 y^{2}(x+\cos x), y(0)=-2
$$

Robert D.

Numerade Educator

Problem 276

For the following problems, find the solution to the initial value problem.
$$
(x-1) y^{\prime}=y-2, y(0)=0
$$

Prathan J.

Numerade Educator

Problem 277

For the following problems, find the solution to the initial value problem.
$$
y^{\prime}=3 y-x+6 x^{2}, y(0)=-1
$$

Robert D.

Numerade Educator

Problem 278

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

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Problem 279

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.
$$
y^{\prime}=\frac{1}{x}+\ln x-y, \text { for } x > 0
$$

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Problem 280

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?
$$
y^{\prime}=-4 y x, y(0)=1
$$

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Problem 281

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?
$$
y^{\prime}=3^{x}-2 y, y(0)=0
$$

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Problem 282

For the following problems, set up and solve the differential equations.
A car drives along a freeway, accelerating according to $a=5 \sin (\pi t), \quad$ where $t$ represents time in minutes. Find the velocity at any time $t, \quad$ assuming the car starts with an initial speed of 60 $\mathrm{mph} .$

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Problem 283

For the following problems, set up and solve the differential equations.
You throw a ball of mass 2 kilograms into the air with an upward velocity of 8 $\mathrm{m} / \mathrm{s}$ . Find exactly the time the ball will remain in the air, assuming that gravity is given by $g=9.8 \mathrm{m} / \mathrm{s}^{2}$

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Problem 284

For the following problems, set up and solve the differential equations.
You drop a ball with a mass of 5 kilograms out an airplane window at a height of 5000 $\mathrm{m} .$ How long does it take for the ball to reach the ground?

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Problem 285

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

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Problem 286

For the following problems, set up and solve the differential equations.
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 $\mathrm{mg}$ to be in the bloodstream at all times, how large should the dose be?

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Problem 287

For the following problems, set up and solve the differential equations.
A 1000 -liter tank contains pure water and a solution of 0.2 kg saltL is pumped into the tank at a rate of 1 $\mathrm{L} /$ min and is drained at the same rate. Solve for total amount of salt in the tank at time $t$ .

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Problem 288

For the following problems, set up and solve the differential equations.
You boil water to make tea. When you pour the water into your teapot, the temperature is $100^{\circ} \mathrm{C}$ . After 5 minutes in your $15^{\circ} \mathrm{C}$ room, the temperature of the tea is $85^{\circ} \mathrm{C}$ . 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 $\left(72^{\circ} \mathrm{C}\right) ?$

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Problem 289

For the following problems, set up and solve the differential equations.
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 $1,998,257 ?$

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Problem 290

For the following problems, set up and solve the differential equations.
Repeat the previous problem but use Gompertz growth model. Which is more accurate?

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