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Calculus for AP

Jon Rogawski & Ray Cannon

Chapter 9

INTRODUCTION TO DIFFERENTIAL EQUATIONS

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Problem 1

Which of the following differential equations are first-order?
(a) $y^{\prime}=x^{2} \quad$ (b) $y^{\prime \prime}=y^{2}$
(c) $\left(y^{\prime}\right)^{3}+y y^{\prime}=\sin x \quad$ (d) $x^{2} y^{\prime}-e^{x} y=\sin y$
(e) $y^{\prime \prime}+3 y^{\prime}=\frac{y}{x} \quad$ (f) $y y^{\prime}+x+y=0$

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Problem 2

Which of the equations in Exercise 1 are linear?

Julia C.
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Problem 3

Verify that the given function is a solution of the differential equation.
$y^{\prime}-8 x=0, \quad y=4 x^{2}$

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Problem 4

Verify that the given function is a solution of the differential equation.
$y y^{\prime}+4 x=0, \quad y=\sqrt{12-4 x^{2}}$

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Problem 5

Verify that the given function is a solution of the differential equation.
$y^{\prime}+4 x y=0, \quad y=25 e^{-2 x^{2}}$

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Problem 6

Verify that the given function is a solution of the differential equation.
$\left(x^{2}-1\right) y^{\prime}+x y=0, \quad y=4\left(x^{2}-1\right)^{-1 / 2}$

Julia C.
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Problem 7

Verify that the given function is a solution of the differential equation.
$y^{\prime \prime}-2 x y^{\prime}+8 y=0, \quad y=4 x^{4}-12 x^{2}+3$

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Problem 8

Verify that the given function is a solution of the differential equation.
$y^{\prime \prime}-2 y^{\prime}+5 y=0, \quad y=e^{x} \sin 2 x$

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Problem 9

Which of the following equations are separable? Write those that are separable in the form $y^{\prime}=f(x) g(y)($ but do not solve).
(a) $x y^{\prime}-9 y^{2}=0 \quad$ (b) $\sqrt{4-x^{2}} y^{\prime}=e^{3 y} \sin x$
(c) $y^{\prime}=x^{2}+y^{2} \quad$ (d) $y^{\prime}=9-y^{2}$

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Problem 10

The following differential equations appear similar but have very different solutions.
$$\frac{d y}{d x}=x, \quad \frac{d y}{d x}=y.$$
Solve both subject to the initial condition $y(1)=2$.

Julia C.
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Problem 11

Consider the differential equation $y^{3} y^{\prime}-9 x^{2}=0$.
(a) Write it as $y^{3} d y=9 x^{2} d x$.
(b) Integrate both sides to obtain $\frac{1}{4} y^{4}=3 x^{3}+C$.
(c) Verify that $y=\left(12 x^{3}+C\right)^{1 / 4}$ is the general solution.
(d) Find the particular solution satisfying $y(1)=2$ .

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Problem 12

Verify that $x^{2} y^{\prime}+e^{-y}=0$ is separable.
(a) Write it as $e^{y} d y=-x^{-2} d x$.
(b) Integrate both sides to obtain $e^{y}=x^{-1}+C$ .
(c) Verify that $y=\ln \left(x^{-1}+C\right)$ is the general solution.
(d) Find the particular solution satisfying $y(2)=4$ .

Julia C.
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Problem 13

Use separation of variables to find the general solution.
$y^{\prime}+4 x y^{2}=0$

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Problem 14

Use separation of variables to find the general solution.
$y^{\prime}+x^{2} y=0$

Julia C.
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Problem 15

Use separation of variables to find the general solution.
$\frac{d y}{d t}-20 t^{4} e^{-y}=0$

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Problem 16

Use separation of variables to find the general solution.
$t^{3} y^{\prime}+4 y^{2}=0$

Julia C.
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Problem 17

Use separation of variables to find the general solution.
$2 y^{\prime}+5 y=4$

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Problem 18

Use separation of variables to find the general solution.
$\frac{d y}{d t}=8 \sqrt{y}$

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Problem 19

Use separation of variables to find the general solution.
$\sqrt{1-x^{2}} y^{\prime}=x y$

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Problem 20

Use separation of variables to find the general solution.
$y^{\prime}=y^{2}\left(1-x^{2}\right)$

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Problem 21

Use separation of variables to find the general solution.
$y y^{\prime}=x$

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Problem 22

Use separation of variables to find the general solution.
$(\ln y) y^{\prime}-t y=0$

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Problem 23

Use separation of variables to find the general solution.
$\frac{d x}{d t}=(t+1)\left(x^{2}+1\right)$

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Problem 24

Use separation of variables to find the general solution.
$\left(1+x^{2}\right) y^{\prime}=x^{3} y$

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Problem 25

Use separation of variables to find the general solution.
$y^{\prime}=x \sec y$

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Problem 26

Use separation of variables to find the general solution.
$\frac{d y}{d \theta}=\tan y$

Julia C.
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Problem 27

Use separation of variables to find the general solution.
$\frac{d y}{d t}=y \tan t$

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Problem 28

Use separation of variables to find the general solution.
$\frac{d x}{d t}=t \tan x$

Julia C.
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Problem 29

Solve the intial value problem.
$y^{\prime}+2 y=0, \quad y(\ln 5)=3$

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Problem 30

Solve the intial value problem.
$y^{\prime}-3 y+12=0, \quad y(2)=1$

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Problem 31

Solve the intial value problem.
$y y^{\prime}=x e^{-y^{2}}, \quad y(0)=-2$

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Problem 32

Solve the intial value problem.
$y^{2} \frac{d y}{d x}=x^{-3}, \quad y(1)=0$

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Problem 33

Solve the intial value problem.
$y^{\prime}=(x-1)(y-2), \quad y(2)=4$

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Problem 34

Solve the intial value problem.
$y^{\prime}=(x-1)(y-2), \quad y(2)=2$

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Problem 35

Solve the intial value problem.
$y^{\prime}=x\left(y^{2}+1\right), \quad y(0)=0$

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Problem 36

Solve the intial value problem.
$(1-t) \frac{d y}{d t}-y=0, \quad y(2)=-4$

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Problem 37

Solve the intial value problem.
$\frac{d y}{d t}=y e^{-t}, \quad y(0)=1$

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Problem 38

Solve the intial value problem.
$\frac{d y}{d t}=t e^{-y}, \quad y(1)=0$

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Problem 39

Solve the intial value problem.
$t^{2} \frac{d y}{d t}-t=1+y+t y, \quad y(1)=0$

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Problem 40

Solve the intial value problem.
$\sqrt{1-x^{2}} y^{\prime}=y^{2}+1, \quad y(0)=0$

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Problem 41

Solve the intial value problem.
$y^{\prime}=\tan y, \quad y(\ln 2)=\frac{\pi}{2}$

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Problem 42

Solve the intial value problem.
$y^{\prime}=y^{2} \sin x, \quad y(\pi)=2$

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Problem 43

Find all values of $a$ such that $y=x^{a}$ is a solution of
$$y^{\prime \prime}-12 x^{-2} y=0$$

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Problem 44

Find all values of $a$ such that $y=e^{a x}$ is a solution of
$$y^{\prime \prime}+4 y^{\prime}-12 y=0$$

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Problem 45

Let $y(t)$ be a solution of $(\cos y+1) \frac{d y}{d t}=2 t$ such that $y(2)=0$.

Show that $\sin y+y=t^{2}+C .$ We cannot solve for $y$ as a function of $t,$ but, assuming that $y(2)=0,$ find the values of $t$ at which $y(t)=\pi$.

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Problem 46

Assuming that $y(6)=\pi / 3,$ find an equation of the tangent line to the graph of $y(t)$ at $(6, \pi / 3)$.

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Problem 47

Use $E q .(4)$ and Torricelli's Law $[E q .(5)]$.

Water leaks through a hole of area 0.002 $\mathrm{m}^{2}$ at the bottom of a cylindrical tank that is filled with water and has height 3 $\mathrm{m}$ and a base of area 10 $\mathrm{m}^{2} .$ How long does it take (a) for half of the water to leak out and (b) for the tank to empty?

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Problem 48

Use $E q .(4)$ and Torricelli's Law $[E q .(5)]$.

At $t=0,$ a conical tank of height 300 $\mathrm{cm}$ and top radius 100 $\mathrm{cm}$ $[$ Figure 7$(\mathrm{A})]$ is filled with water. Water leaks through a hole in the bottom of area 3 $\mathrm{cm}^{2} .$ Let $y(t)$ be the water level at time $t .$
(a) Show that the tank's cross-sectional area at height $y$ is $A(y)=$ $\frac{\pi}{9} y^{2}$ .
(b) Find and solve the differential equation satisfed by $y(t)$
(c) How long does it take for the tank to empty?

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Problem 49

Use $E q .(4)$ and Torricelli's Law $[E q .(5)]$.

The tank in Figure 7$(\mathrm{B})$ is a cylinder of radius 4 $\mathrm{m}$ and height 15 $\mathrm{m} .$ Assume that the tank is half-filled with water and that water leaks through a hole in the bottom of area $B=0.001 \mathrm{m}^{2} .$ Determine the water level $y(t)$ and the time $t_{e}$ when the tank is empty.

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Problem 50

Use $E q .(4)$ and Torricelli's Law $[E q .(5)]$.

A tank has the shape of the parabola $y=x^{2}$, revolved around the $y$-axis. Water leaks from a hole of area $B=0.0005 \mathrm{m}^{2}$ at the bottom of the tank. Let $y(t)$ be the water level at time $t .$ How long does it take for the tank to empty if it is initially filled to height $y_{0}=1 \mathrm{m}$ .

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Problem 51

Use $E q .(4)$ and Torricelli's Law $[E q .(5)]$.

Atankhas the shape of the parabola $y=a x^{2}$ (where $a$ is a constant) revolved around the $y$-axis. Water drains from a hole of area $B \mathrm{m}^{2}$ at the bottom of the tank.
(a) Show that the water level at time $t$ is
$$y(t)=\left(y_{0}^{3 / 2}-\frac{3 a B \sqrt{2 g}}{2 \pi} t\right)^{2 / 3}$$
where $y_{0}$ is the water level at time $t=0$.
(b) Show that if the total volume of water in the tank has volume $V$ at time $t=0,$ then $y_{0}=\sqrt{2 a V / \pi} .$ Hint: Compute the volume of the tank as a volume of rotation.
(c) Show that the tank is empty at time
$$t_{e}=\left(\frac{2}{3 B \sqrt{g}}\right)\left(\frac{2 \pi V^{3}}{a}\right)^{1 / 4}$$
We see that for fixed initial water volume $V,$ the time $t_{e}$ is proportional to $a^{-1 / 4} .$ A large value of $a$ corresponds to a tall thin tank. Such a tank drains more quickly than a short wide tank of the same initial volume.

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Problem 52

Use $E q .(4)$ and Torricelli's Law $[E q .(5)]$.

A cylindrical tank filled with water has height $h$ and a base of area $A .$ Water leaks through a hole in the bottom of area $B$.
(a) Show that the time required for the tank to empty is proportional to $A \sqrt{h} / B .$
(b) Show that the emptying time is proportional to $V h^{-1 / 2},$ where $V$ is the volume of the tank.
(c) Two tanks have the same volume and a hole of the same size, but they have different heights and bases. Which tank empties first: the taller or the shorter tank?

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Problem 53

Figure 8 shows a circuit consisting of a resistor of $R$ ohms, a capacitor of $C$ farads, and a battery of voltage $V .$ When the circuit is completed, the amount of charge $q(t)$ (in coulombs) on the plates of the capacitor varies according to the differential equation $(t$ in seconds).
$$R \frac{d q}{d t}+\frac{1}{C} q=V$$
where $R, C,$ and $V$ are constants.
(a) Solve for $q(t),$ assuming that $q(0)=0$.
(b) Show that
$$\lim _{t \rightarrow \infty} q(t)=C V.$$
(c) Show that the capacitor charges to approximately 63$\%$ of its final value $C V$ after a time period of length $\tau=R C(\tau$ is called the time constant of the capacitor).

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Problem 54

Assume in the circuit of Figure 8 that $R=200 \Omega, C=0.02 \mathrm{F}$ and $V=12 \mathrm{V}$ . How many seconds does it take for the charge on the capacitor plates to reach half of its limiting value?

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Problem 55

According to one hypothesis, the growth rate $d V / d t$ of a cell's volume $V$ is proportional to its surface area $A .$ since $V$ has cubic units such as $\mathrm{cm}^{3}$ and $A$ has square units such as $\mathrm{cm}^{2},$ we may assume roughly that $A \propto V^{2 / 3},$ and hence $d V / d t=k V^{2 / 3}$ for some constant $k .$ If this hypothesis is correct, which dependence of volume on time would we expect to see (again, roughly speaking) in the laboratory?
(a) Linear $\quad$ (b) Quadratic $\quad$ (c) Cubic

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Problem 56

We might also guess that the volume $V$ of a melting snowball decreases at a rate proportional to its surface area. Argue as in Exercise 55 to find a differential equation satisfied by $V$ . Suppose the snowball has volume 1000 $\mathrm{cm}^{3}$ and that it loses half of its volume after 5 min. According to this model, when will the snowball disappear?

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Problem 57

In general, $(f g)^{\prime}$ is not equal to $f^{\prime} g^{\prime},$ but let $f(x)=e^{3 x}$ and find a function $g(x)$ such that $(f g)^{\prime}=f^{\prime} g^{\prime} .$ Do the same for $f(x)=x$.

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Problem 58

A boy standing at point $B$ on a dock holds a rope of length $\ell$ attached to a boat at point $A[$ Figure 9$(\mathrm{A})] .$ As the boy walks along the dock, holding the rope taut, the boat moves along a curve called a tractrix (from the Latin tractus, meaning "to pull"). The segment from a point $P$ on the curve to the $x$-axis along the tangent line has constant length $\ell .$ Let $y=f(x)$ be the equation of the tractrix.
(a) Show that $y^{2}+\left(y / y^{\prime}\right)^{2}=\ell^{2}$ and conclude $y^{\prime}=-\frac{y}{\sqrt{\ell^{2}-y^{2}}}.$ Why must we choose the negative square root?
(b) Prove that the tractrix is the graph of
$$x=\ell \ln \left(\frac{\ell+\sqrt{\ell^{2}-y^{2}}}{y}\right)-\sqrt{\ell^{2}-y^{2}}$$

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Problem 59

Show that the differential equations $y^{\prime}=3 y / x$ and $y^{\prime}=-x / 3 y$ define orthogonal families of curves; that is, the graphs of solutions to the first equation intersect the graphs of the solutions to the second equation in right angles (Figure 10$) .$ Find these curves explicitly.

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Problem 60

Find the family of curves satisfying $y^{\prime}=x / y$ and sketch several members of the family. Then find the differential equation for the orthogonal family (see Exercise $59 ),$ find its general solution, and add some members of this orthogonal family to your plot.

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Problem 61

A 50 -kg model rocket lifts off by expelling fuel at a rate of $k=4.75 \mathrm{kg} / \mathrm{s}$ for 10 $\mathrm{s}$ . The fuel leaves the end of the rocket with an exhaust velocity of $b=100 \mathrm{m} / \mathrm{s} .$ Let $m(t)$ be the mass of the rocket at time $t .$ From the law of conservation of momentum, we find the following differential equation for the rocket's velocity $v(t)($ in meters per second):
$$m(t) v^{\prime}(t)=-9.8 m(t)+b \frac{d m}{d t}$$
(a) Show that $m(t)=50-4.75 t \mathrm{kg}$ .
(b) Solve for $v(t)$ and compute the rocket's velocity at rocket burnout (after 10 $\mathrm{s}$ ).

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Problem 62

Let $v(t)$ be the velocity of an object of mass $m$ in free fall near the earth's surface. If we assume that air resistance is proportional to $v^{2}$ then $v$ satisfies the differential equation $m \frac{d v}{d t}=-g+k v^{2}$ for some constant $k>0$.
(a) $\operatorname{Set} \alpha=(g / k)^{1 / 2}$ and rewrite the differential equation as
$$\frac{d v}{d t}=-\frac{k}{m}\left(\alpha^{2}-v^{2}\right)$$
Then solve using separation of variables with initial condition $v(0)=0$.
(b) Show that the terminal velocity $$\lim _{t \rightarrow \infty} v(t)$$ is equal to $-\alpha$

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Problem 63

If a bucket of water spins about a vertical axis with constant angular velocity $\omega($ in radians per second), the water climbs up the side of the bucket until it reaches an equilibrium position (Figure 11). Two forces act on a particle located at a distance $x$ from the vertical axis: the gravitational force $-m g$ acting downward and the force of the bucket on the particle (transmitted indirectly through the liquid) in the direction perpendicular to the surface of the water. These two forces must combine to supply a centripetal force $m \omega^{2} x,$ and this occurs if the diagonal of the rectangle in Figure 11 is normal to the water's surface (that is, perpendicular to the tangent line). Prove that if $y=f(x)$ is the equation of the curve obtained by taking a vertical cross section through the axis, then $-1 / y^{\prime}=-g /\left(\omega^{2} x\right) .$ Show that $y=f(x)$ is a parabola.

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Problem 64

In Section 6.2, we computed the volume $V$ of a solid as the integral of cross-sectional area. Explain this formula in terms of differential equations. Let $V(y)$ be the volume of the solid up to height $y,$ and let $A(y)$ be the cross-sectional area at height $y$ as in Figure 12.
(a) Explain the following approximation for small $\Delta y :$
$$V(y+\Delta y)-V(y) \approx A(y) \Delta y$$
(b) Use Eq. (8) to justify the differential equation $d V / d y=A(y)$ . Then derive the formula
$$V=\int_{a}^{b} A(y) d y$$

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Problem 65

A basic theorem states that a linear differential equation of order $n$ has a general solution that depends on $n$ arbitrary constants. There are, however, nonlinear exceptions.
(a) Show that $\left(y^{\prime}\right)^{2}+y^{2}=0$ is a first-order equation with only one solution $y=0 .$
(b) Show that $\left(y^{\prime}\right)^{2}+y^{2}+1=0$ is a first-order equation with no solutions.

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Problem 66

Show that $y=C e^{r x}$ is a solution of $y^{\prime \prime}+a y^{\prime}+b y=0$ if and only if $r$ is a root of $P(r)=r^{2}+a r+b .$ Then verify directly that $y=C_{1} e^{3 x}+C_{2} e^{-x}$ is a solution of $y^{\prime \prime}-2 y^{\prime}-3 y=0$ for any constants $C_{1}, C_{2} .$

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Problem 67

A spherical tank of radius $R$ is half-filled with water. Suppose that water leaks through a hole in the bottom of area $B$ . Let $y(t)$ be the water level at time $t($ seconds $) .$
(a) Show that $\frac{d y}{d t}=\frac{-8 B \sqrt{y}}{\pi\left(2 R y-y^{2}\right)}$.
(b) Show that for some constant $C$,
$$\frac{\pi}{60 B}\left(10 R y^{3 / 2}-3 y^{5 / 2}\right)=C-t$$
(c) Use the initial condition $y(0)=R$ to compute $C,$ and show that $C=t_{e},$ the time at which the tank is empty.
(d) Show that $t_{e}$ is proportional to $R^{5 / 2}$ and inversely proportional to $B .$

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