Calculus: Early Transcendentals

James Stewart

Chapter 9 Differential Equations - all with Video Answers

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Section 5

Linear Equations

00:31

Problem 1

Determine whether the differential equation is linear.
$ y' + x \sqrt y = x^2 $

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00:38

Problem 2

Determine whether the differential equation is linear.
$ y' - x = y \tan x $

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00:57

Problem 3

Determine whether the differential equation is linear.
$ ue^t = t + \sqrt t \frac {du}{dt} $

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00:33

Problem 4

Determine whether the differential equation is linear.
$ \frac {dR}{dt} + t \cos R = e^{-t} $

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01:10

Problem 5

Solve the differential equation.
$ y' + y = 1 $

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01:20

Problem 6

Solve the differential equation.
$ y' - y = e^x $

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01:23

Problem 7

Solve the differential equation.
$ y' = x - y $

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02:04

Problem 8

Solve the differential equation.
$ 4x^3y + x^4y' = \sin^3x $

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01:07

Problem 9

Solve the differential equation.
$ xy' + y = \sqrt x $

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01:39

Problem 10

Solve the differential equation.
$ 2xy' + y = 2 \sqrt x $

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01:40

Problem 11

Solve the differential equation.
$ xy' - 2y = x^2, x > 0 $

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04:34

Problem 12

Solve the differential equation.
$ y' + 2xy = 1 $

Yaping Yuan

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02:06

Problem 13

Solve the differential equation.
$ t^2 \frac {dy}{dt} + 3ty = \sqrt {1 + t^2}, t > 0 $

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02:07

Problem 14

Solve the differential equation.
$ t \ln t \frac {dr}{dt} + r = te^t $

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01:27

Problem 15

Solve the initial-value problem.
$ x^2y' + 2xy = \ln x, y(1) = 2 $

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01:50

Problem 16

Solve the initial-value problem.
$ t^3 \frac{du}{dt} + 3t^2y = \cos t, y(\pi) = 0 $

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01:50

Problem 17

Solve the initial-value problem.
$ t \frac {du}{dt} = t^2 + 3u, t > 0, u(2) = 4 $

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01:18

Problem 18

Solve the initial-value problem.
$ xy' + y = x \ln x, y(1) = 0 $

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01:46

Problem 19

Solve the initial-value problem.
$ xy' = y + x^2 \sin x, y(\pi) = 0 $

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03:06

Problem 20

Solve the initial-value problem.
$ (x^2 + 1) \frac {dy}{dx} + 3x(y -1) = 0, y(0) = 2 $

Amrita Bhasin

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03:00

Problem 21

Solve the differential equation and use a calculator to graph several members of the family of solutions. How does the solution curve change as $ C $ varies?
$ xy' + 2y = e^x $

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05:34

Problem 22

Solve the differential equation and use a calculator to graph several members of the family of solutions. How does the solution curve change as $ C $ varies?
$ xy' = x^2 + 2y $

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01:10

A Bernoulli differential equation (named after James Bernoulli) is of the form
$ \frac {dy}{dx} + P(x)y = Q(x)y^n $
Observe that, if $ n = 0 $ or 1, the Bernoulli equation is linear. For other values of $ n, $ show that the substitution $ u = y^{1-n} $ transforms the Bernoulli equation into the linear equation
$ \frac {du}{dx} + ( 1 - n)P(x)u = (1 - n)Q(x) $

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03:04

Problem 24

Use the method of Exercise 23 to solve the differential equation.
$ xy' + y = -xy^2 $

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03:37

Problem 25

Use the method of Exercise 23 to solve the differential equation.
$ y' + \frac {2}{x}y = \frac {y^3}{x^2} $

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02:08

Problem 26

Solve the second-order equation $ xy'' + 2y' = 12x^2 $ by making the substitution $ u = y'. $

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05:37

Problem 27

In the circuit shown in Figure 4, a battery supplies a constant voltage 40 V, the inductance is 2 H, the resistance is $ 10 \Omega, $ and $ I(0) = 0. $
(a) Find $ I(t). $
(b) Find the current after 0.1 seconds.

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09:55

Problem 28

In the circuit shown in Figure 4, a generator supplies a voltage of $ E(t) = 40 \sin 60t $ volts, the inductance is 1 H, the resistance is $ 20 \Omega, $ and $ I(0) = 1 A. $
(a) Find $ I(t). $
(b) Use a graphing device to draw the graph of the current function.

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07:16

Problem 29

The figure shows a circuit containing an electromotive force, a capacitor with a capacitance of $ C $ farads $ (F), $ and a resistor with resistance of $ R $ ohms $ (\Omega). $ The voltage drop across is $ Q/C, $ where $ Q $ is the charge (in coulombs), so in this case Kirchhoff's Law gives
$ RI + \frac {Q}{C} = E(t) $
But $ I = dQ/dt $ (see Examples 3.7.3), so we have
$ R \frac {dQ}{dt} + \frac {1}{C}Q = E(t) $
Suppose the resistance is $ 5 \Omega, $ the capacitance is 0.05 F, a battery gives a constant voltage of 60 V, and the initial charge is $ Q(0) = 0 C. $ Find the charge and the current at time $ t. $

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09:46

Problem 30

In the circuit of Exercise 29, $ R = 2 \Omega, C = 0.01 F, Q(0) = 0, $ and $ E(t) = 10 \sin 60t. $ Find the charge and the current at time $ t. $

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05:30

Problem 31

Let $ P(t) $ be the performance level of someone learning a skill as a function of the training time $ t. $ The graph of $ P $ is called a learning curve. In Exercise 9.1.15 we proposed the differential equation
$ \frac {dP}{dt} = k[M - P(t)] $
as a reasonable model for learning, where $ k $ is a positive constant. Solve it as a linear differential equation and use your solution to graph the learning curve.

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08:09

Problem 32

Two new workers were hired for an assembly line. Jim processed 25 units during first hour and 45 units during the second hour. Mark processed 35 units during the first hour and 50 units the second hour. Using the model of Exercise 31 and assuming that $ P(0) = 0, $ estimate the maximum number of units per hour that each worker is capable of processing.

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10:38

Problem 33

In Section 9.3 we looked at mixing problems in which the volume of fluid remained constant and saw that such problems give rise to separable differentiable equations. (See Example 6 in that section.) If the rates of flow into and out of the system are different, then the volume is not constant and the resulting differential equation is linear but not separable.
A tank contains 100 L of water. A solution with a salt concentration of 0.4 kg/L is added at a rate of 5 L/min. The solution is kept mixed and is drained from the tank at a rate of 3 L/min. If $ y(t) $ is the amount of salt (in kilograms) after $ t $ minutes, show that $ y $ satisfies the differential equation
$ \frac {dy}{dt} = 2 - \frac {3y}{100 + 2t} $
Solve this equation and find the concentration after 20 minutes.

Charles Carter

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Problem 34

A tank with a capacity of 400 L is full of a mixture of water and chlorine with a concentration of 0.05 g of chlorine per liter. In order to reduce the concentration of chlorine, fresh water is pumped into the tank at a rate of 4 L/s. The mixture is kept stirred and is pumped out at a rate of 10 L/s. Find the amount of chlorine in the tank as a function of time.

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12:25

Problem 35

An object with mass $ m $ is dropped from rest and we assume that the air resistance is proportional to the speed of the object. If $ s(t) $ is the distance dropped after $ t $ seconds, then the speed is $ v = s'(t) $ and the acceleration is $ a = v'(t). $ If $ g $ is the acceleration due to gravity, them the downward force on the object is $ mg - cv, $ where $ c $ is a positive constant, and Newton's Second Law gives
$ m \frac {dv}{dt} = mg - cv $
(a) Solve this as a linear equation to show that
$ v = \frac {mg}{c} (1 - e^{-ct/m}) $
(b) What is the limiting velocity?
(c) Find the distance the object has fallen after $ t $ seconds.

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05:09

Problem 36

If we ignore air resistance, we can conclude that heavier objects fall no faster than lighter objects. But if we take air resistance into account, our conclusion changes. Use the expression for the velocity of a falling object in Exercise 35(a) to find $ dv/dm $ and show that heavier objects $ do $ fall faster than lighter ones.

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05:45

Problem 37

(a) Show that the substitution $ z = 1/P $ transforms the logistic differential equation $ P' = kP(1 - P/M) $ into the linear differential equation
$ z' + kz = \frac {k}{M} $
(b) Solve the linear differential equation in part (a) and thus obtain an expression for $ P(t). $ Compare with Equation 9.4.7.

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Problem 38

To account for seasonal variation in the logistic differential equation, we could allow $ k $ and $ M $ to be functions of $ t: $
$ \frac {dP}{dt} = k(t)P (1 - \frac {P}{M(t)}) $
(a) Verify that the substitution $ z = 1/P $ transform this equation into the linear equation
$ \frac {dz}{dt} + k(t)z = \frac {k(t)}{M(t)} $
(b) Write an expression for the solution of the equation in part (a) and use it to show that if the carrying capacity $ M $ is constant, then
$ P(t) = \frac {M}{1 + CMe^{-\int k(t) dt}} $
Deduce that if $ \int^\infty_0 k(t) dt = \infty, $ then $ \lim_{t \to \infty} P(t) = M. $ [This will be true if $ k(t) = k_0 + a \cos bt $ with $ k_0 > 0, $ which describes a positive instrinsic growth rate with a periodic seasonal variation.]
(c) If $ k $ is constant but $ M $ varies, show that
$ z(t) = e^{-kt} \int^t_0 \frac {ke^{ks}}{M(s)} ds + Ce^{-kt} $
and use 1'Hospital's Rule to deduce that if $ M(t) $ has a limit as $ t \to \infty, $ then $ P(t) $ has the same limit.

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