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True or False First-order linear differential equations have the form \(\dfrac{dy}{dx}+P(x)y=Q(x)\), where the functions \(P \) and \(Q\) are continuous on their domains.

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True or False The first-order linear differential equation \(\dfrac{dy}{dx}+P(x)y=0\) is separable.

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True or False Multiplying a first-order linear differential equation \(\dfrac{dy}{dx}+P(x)y=Q(x)\), where \(Q( x) \neq 0\), by the integrating factor \(e^{\int Q( x)\,dx}\) results in a separable differential equation.

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True or False To solve a Bernoulli equation, \(\dfrac{ dy}{dx}+P(x)y=Q(x)y^{n}\), \(n\neq 0\), \(n\neq 1\), the first step is to multiply by \(y^{n}\).

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\(\dfrac{dy}{dx}+2xy=0\)

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\(\dfrac{dy}{dx}+\dfrac{1}{x}y=0\)

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\(\dfrac{dy}{dx}+\dfrac{1}{x}y=3x\)

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\(\dfrac{dr}{d\theta }+\dfrac{4r}{\theta }=\theta \)

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\(\dfrac{dy}{dx}+\dfrac{y}{x}=x^{2}\)

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\(\dfrac{dy}{dx}-\dfrac{2y}{x+1}=3(x+1)^{2}\)

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\(\dfrac{dy}{dx}-2y=e^{-x}\)

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\(\dfrac{dy}{dx}-\dfrac{y}{x}=x^{3/2}\)

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\(\dfrac{dy}{dx}+\dfrac{2y}{x}=x^{2}+1\)

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\(\dfrac{dy}{dx}+2xy=2x\)

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\(\dfrac{dy}{dx}+e^{x}y=e^{x}\)

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\(\dfrac{dy}{dx}+e^{-x}y=e^{-x}\)

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\(\dfrac{dy}{dx}+y\tan x=\cos x\)

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\(\dfrac{dy}{dx}-y\csc x=\sin 2x\)

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\(\dfrac{dy}{dx}+y\cot x=\csc ^{2}x\)

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\(\dfrac{dy}{dx}+y\tan x=\cos ^{2}x\)

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\(\dfrac{dy}{dx}+x^{-1}y=3x^{2}y^{3}\)

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\(\dfrac{dy}{dx}+x^{-1}y=\dfrac{2}{3}x^{2}y^{4}\)

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\(\dfrac{dy}{dx}+2xy=xy^{4}\)

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\(\dfrac{dy}{dx}+\dfrac{1}{x}y=3xy^{3}\)

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\(\dfrac{dy}{dx}+\dfrac{1}{x}y=3xy^{1/3}\)

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\(\dfrac{dy}{dx}+\dfrac{4y}{x}=xy^{1/2}\)

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\(x\dfrac{dy}{dx}-y=x^{2}e^{x}\)

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\(\dfrac{dy}{dx}=\dfrac{y}{x-y^{2}}\)

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\(dx+(2x-y^{2})\,dy=0\)

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\((2x+y)\,dx-xdy=0\)

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\(\cos x\dfrac{dy}{dx}+y=\sec x\)

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\((1+x^{2})\dfrac{dy}{dx}+xy=x^{3}\)

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\(dy+y\,dx=2xy^{2}e^{x}\,dx\)

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\(dx+2xy^{-1}\,dy=2x^{2}y^{2}\,dy\)

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\(2\dfrac{dy}{dx}-yx^{-1}=5x^{3}y^{3}\)

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\(dx-2xy\,dy=6x^{3}y^{2}e^{-2y^{2}}\,dy\)

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\((x^{2}+2y^{2})\dfrac{dx}{dy}+xy=0\)

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\((3x^{2}y^{2}-e^{y}x^{4}y)\dfrac{dy}{dx}=2xy^{3}\)

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\(\dfrac{dy}{dx}+y=e^{-x};\quad y=5\) when \(x=0\)

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\(\dfrac{dy}{dx}+\dfrac{2y}{x}=\dfrac{4}{x};\quad y=6\) when \(x=1\)

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\(\dfrac{dy}{dx}+\dfrac{y}{x}=e^{x};\quad y=e^{-1}\) when \(x=-1\)

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\(\dfrac{dy}{dx}+y\cot x=2\cos x;\quad y=3\) when \(x= \dfrac{\pi }{2}\)

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\(\dfrac{dy}{dx}+\dfrac{3y}{x}=\sin x;\quad y( \pi) =0\)

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\(x\dfrac{dy}{dx}+2y=\dfrac{5\sin x}{x};\quad y ( \pi ) =0\)

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Free Fall Using \(v(t)=-\dfrac{mg}{k}+\left( v_{0}+\dfrac{mg }{k}\right) e^{-(k/m)t}\), find the position \(s(t)\) of a freely falling object at any time \(t\). Assume that its initial position is \(s(0)=s_{0}\).

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Free Fall An object of mass \(m=\) \(2\;\rm{kg}\) is dropped from rest from a height of \(2000\;\rm{m}\). As it falls, the air resistance is equal to \(\dfrac{1}{2}v\), where \(v\) is the velocity measured in meters per second.

1. Find the velocity \(v\) and the distance \(s\) the object has fallen at time \(t\).
2. What is the limiting velocity of the object?
3. With what velocity does the object strike Earth?

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Velocity of a Skydiver A skydiver and her parachute together weigh \(160\;\rm{lb}\). She free falls from rest from a height of \(10,000\;\rm{ft}\) for \(5\;\rm{seconds}\). Assume that there is no air resistance during the fall. After her parachute opens, the air resistance is four times her velocity \(v\).

1. How fast will the skydiver be falling \(4\;\rm{seconds}\) after the parachute opens?
2. How long will it take for the skydiver to land on the ground?
3. What is her velocity \(v\) when she lands?

(Hint: There are two distinct differential equations that govern the velocity and position of the skydiver: one for the free fall period and the other for the period after the parachute opens.)

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Spread of a Rumor A rumor spreads through a population of \(5000\) people at a rate that is proportional to the product of the number of people who have heard the rumor and the number who have not heard it. Suppose that \(100\) people initiated the rumor and \(500\) have heard it after \(3\) days.

1. Write a differential equation that models the rate \(\dfrac{dy}{dt}\) at which the rumor spreads, where \(y=y( t)\), is the number of people who have heard the rumor after \(t\) days.
2. Solve the differential equation from (a).
3. How many people will have heard the rumor after 8 days?
4. How long will it take for half the people to hear the rumor?

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Mixtures A tank initially contains \(100\) liters (L) of pure water. Starting at time \(t=0\), brine containing \(3\;\rm{kg}\) of salt per liter flows into the tank at the rate of \(8\) L\(/\!\rm{min}\). The mixture is kept uniform by stirring, and the well-stirred mixture flows out of the tank at the same rate as it flows in. How much salt is in the tank after \(5\;\rm{min}\)? How much salt is in the tank after a very long time?

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Flu Epidemic A flu virus is spreading through a college campus of \(10,000\) students at a rate that is proportional to the product of the number of infected students and the number of noninfected students. Assume \(10\) students were infected initially and \(200\) students are infected after \(10\) days.

1. Write a differential equation that models the rate \(\dfrac{dy}{dt}\) of infection with respect to time, where \(y=y(t)\) is the number of students infected at time \(t\) in days.
2. Solve the differential equation from (a).
3. How many students will be infected after \(5\) days?
4. How long will it take for \(75\%\) of the students to be infected?
5. Graph \(y=y(t)\) for the first 40 days of the flu epidemic.

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Urban Planning The developers of a planned community assume that the population \(P(t)\) of the community will be governed by the logistic equation \[ \frac{dP}{dt}=P(10^{-2}-10^{-6}P) \]

where \(t\) is measured in months.

1. Solve the differential equation if the initial population is estimated to be \(P(0)=1500\).
2. What is the limiting size of the population?
3. When will the population equal one-half of the limiting value?
4. Graph the predicted population for \(0\leq t\leq 60\) (5 years).

1088

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Kirchhoff’s Law The basic equation governing the amount of current \(I\) (in amperes) in a simple \(RL\) circuit consisting of a resistance \(R\) (in ohms), an inductance \(L\) (in henrys), and an electromotive force \(E\) (in volts) is \[ \begin{equation*} \frac{dI}{dt}+\frac{R}{L}I=\frac{E}{L} \end{equation*} \]

where \(t\) is the time in seconds. Solve the differential equation, assuming that \(E,R\), and \(L\) are constants and \(I=0\) when \(t=0\).

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Electrical Charge The equation governing the amount of electrical charge \(q\) (in coulombs) of an \(RC\) circuit consisting of a resistance \(R\) (in ohms), a capacitance \(C\) (in farads), an electromotive force \(E\), and no inductance is \[ \frac{dq}{dt}+\frac{1}{RC}q=\frac{E}{R} \]

where \(t\) is the time in seconds. Solve the differential equation, assuming \(E,R\), and \(C\) are constants and \(q=0\) when \(t=0\).

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Population Growth Consider modeling the growth of a population of fish in a pond as a case of inhibited growth, where the rate of growth of the number of fish is given by the logistic equation \(\dfrac{dN}{dt} =kN( N_{\max }-N)\), where \(t\) is the time in years, \(N\) is the number of fish at time \(t\), \(N_{\max }\) is the maximum number of fish the pond can support, and \(k\) is a positive constant.

1. Solve the differential equation.
2. If the population doubles after \(5\) years, the initial population is \(20\) fish, and \(N_{\max }=400\) fish, express \(N\) as a function of \(t\).
3. What will the population be after 43 years?
4. Show that the population is growing the fastest when \(N=\dfrac{N_{\max }}{2}\).
5. Graph \(N=N( t)\).

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Population Growth A second model for inhibited growth is called the Gompertz equation, given by the solution to the differential equation \(\dfrac{dN}{dt}=pN\ln \left( \dfrac{N_{\max }}{N} \right)\), where \(N\) is the population at time \(t\) in years, \(N_{\max }\) is the maximum population the environment can support, and \(p\) is a positive constant. Suppose the population of fish in the pond from Problem 54 follows a Gompertz equation.

1. Solve the differential equation.
2. If the population doubles after \(5\) years, the initial population is \(20\) fish, and \(N_{\max }=400\) fish, express \(N\) as a function of \(t\).
3. What will the population be after 43 years?
4. Show that the population is growing the fastest when \(N=\dfrac{N_{\max } }{e}\).
5. Graph \(N=N( t), 0\le t\le 100\).

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Rate of Growth If all members of a population are in contact with every other member, the rate of growth of a fad at any time \(t\) among the population is proportional to the product \(xy\), where \(y\) is the number who have adopted the fad and \(x\) is the number who have not adopted the fad at time \(t\). Suppose that on a certain day \((t=0)\), two members from a club of \(30\) members begin wearing a new style of clothing and three members have adopted the clothing style after 2 days.

1. Express the number of members \(y\) in the club who have adopted the style as a function of \(t\) days.
2. In approximately how many days will half of the club members have adopted the style?

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Consider the logistic differential equation \[ \begin{equation*} \dfrac{dy}{dt}=ky(M-y) \end{equation*} \]

Show that \(\dfrac{dy}{dt}\) is increasing if \(y\,{<}\,\dfrac{M}{2}\) and is decreasing if \(y\,{>}\,\dfrac{M}{2}\). From this, it follows that the growth rate is a maximum when \(y=\dfrac{M}{2}\).

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Solve the logistic differential equation \(\dfrac{dy}{dt} =ky\left( M-y\right)\) by separating the variables and integrating the resulting rational function using partial fractions. Compare the answer obtained with the “Bernoulli solution.”

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Air Quality A room \(150\;\rm{ft}\) by \(50\;\rm{ft}\) by \(20\;\rm{ft}\) receives fresh air at the rate of \(5000\;\rm{ft}^{3}\)/\(\rm{min}\). If the fresh air contains \(0.04\%\) carbon dioxide and the air in the room initially contained \(0.3\%\) carbon dioxide, find the percentage of carbon dioxide after \(1\;\rm{h}\). What is the percentage after \(2\;\rm{h}\)? Assume the mixed air leaves the room at the rate of \(5000\;\rm{ft}^{3}\)/\(\rm{min}\).

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Renewing Currency A nation’s federal bank has \(\$3\) billion of paper currency in circulation. Each day about \(\$10\) million comes into the bank and the same amount is paid out. The federal reserve decides to issue new currency, and whenever the old-style currency comes into the bank, it is destroyed and replaced by the new currency. How long will it take for the currency in circulation to become \(95\%\) new?

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Kirchhoff’s Law The equation governing the amount of current \(I\) (in amperes) in an \(RL\) circuit consisting of a resistance \(R\) (in ohms), an inductance \(L\) (in henrys), and an electromotive force \(E_{0}\sin (\omega t) \) volts is given by Kirchhoff’s Second Law: \[ \begin{equation*} L\frac{dI}{dt}+RI=E_{0}\sin (\omega t) \qquad \omega > 0 \end{equation*} \]

Find \(I\) as a function of \(t\) if \(I=I_{0}\) when \(t=0\). Here, \(R,I,E_0\) and \(\omega\) are constants.