Chapter 1

Solve the differential equation for Newton's law of cooling by viewing it as a first-order linear differential equation.

Determine all values of the constant \(r\) such that the given function solves the given differential equation. $$y(x)=x^{r}, \quad x^{2} y^{\prime \prime}+x y^{\prime}-y=0$$.

Sketch the slope field and some representative solution curves for the given differential equation. $$y^{\prime}=x+y$$

An object is released from rest at a height of 100 meters above the ground. Neglecting frictional forces, the subsequent motion is governed by the initial-value problem $$ \frac{d^{2} y}{d t^{2}}=g, \quad y(0)=0, \quad \frac{d y}{d t}(0)=0 $$where \(y(t)\) denotes the displacement of the object from its initial position at time \(t\). Solve this initial-value problem and use your solution to determine the time when the object hits the ground.

Solve the given initial-value problem. $$\frac{d y}{d x}=\frac{2(2 y-x)}{x+y}, \quad y(0)=2$$

Determine an integrating factor for the given differential equation, and hence find the general solution. $$x^{2} y d x+y\left(x^{3}+e^{-3 y} \sin y\right) d y=0$$

Determine all values of the constant \(r\) such that the given function solves the given differential equation. $$y(x)=x^{r}, \quad x^{2} y^{\prime \prime}+5 x y^{\prime}+4 y=0$$.

Sketch the slope field and some representative solution curves for the given differential equation. $$y^{\prime}=x+y$$

A five-foot-tall boy tosses a tennis ball straight up from the level of the top of his head. Neglecting frictional forces, the subsequent motion is governed by the differential equation $$ \frac{d^{2} y}{d t^{2}}=g $$ If the object hits the ground 8 seconds after the boy releases it, find (a) the time when the tennis ball reaches its maximum height. (b) the maximum height of the tennis ball.

Solve the given initial-value problem. $$\frac{d y}{d x}=\frac{2 x-y}{x+4 y}, \quad y(1)=1$$

Determine which of the five types of differential equations we have studied the given equation falls into (see Table \(1.12 .1),\) and use an appropriate technique to find the general solution. $$e^{2 x+y} d y-e^{x-y} d x=0$$

Determine an integrating factor for the given differential equation, and hence find the general solution. $$(x y-1) d x+x^{2} d y=0$$

Between 8 a.m. and 12 p.m. on a hot summer day, the temperature rose at a rate of \(10^{\circ} \mathrm{F}\) per hour from an initial temperature of \(65^{\circ} \mathrm{F}\). At 9 a.m., the temperature of an object was measured to be \(35^{\circ} \mathrm{F}\) and was, at that time, increasing at a rate of \(5^{\circ} \mathrm{F}\) per hour. Show that the temperature of the object at time \(t\) was $$T(t)=10 t-15+40 e^{(1-t) / 8}, \quad 0 \leq t \leq 4$$

At \(2 \mathrm{p} . \mathrm{m}\). on a cool \(\left(34^{\circ} \mathrm{F}\right)\) afternoon in March, Sherlock Holmes measured the temperature of a dead body to be \(38^{\circ} \mathrm{F}\). One hour later, the temperature was \(36^{\circ} \mathrm{F}\). After a quick calculation using Newton's law of cooling, and taking the normal temperature of a living body to be \(98^{\circ} \mathrm{F},\) Holmes concluded that the time of death was 10 a.m. Was Holmes right?

When \(N\) is a positive integer, the Legendre equation $$\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+N(N+1) y=0$$ with \(-1

Sketch the slope field and some representative solution curves for the given differential equation. $$y^{\prime}=-4 x / y$$

A pyrotechnic rocket is to be launched vertically upwards from the ground. For optimal viewing, the rocket should reach a maximum height of 90 meters above the ground. Ignore frictional forces. (a) How fast must the rocket be launched in order to achieve optimal viewing? (b) Assuming the rocket is launched with the speed determined in part (a), how long after the rocket is launched will it reach its maximum height?

Solve the given initial-value problem. $$\frac{d y}{d x}=\frac{y-\sqrt{x^{2}+y^{2}}}{x}, \quad y(3)=4$$

Determine which of the five types of differential equations we have studied the given equation falls into (see Table \(1.12 .1),\) and use an appropriate technique to find the general solution. $$y^{\prime}+y \cot x=\sec x$$

Determine an integrating factor for the given differential equation, and hence find the general solution. $$\frac{d y}{d x}+\frac{2 x}{1+x^{2}} y=\frac{1}{\left(1+x^{2}\right)^{2}}$$

It is known that a certain object has constant of proportionality \(k=1 / 40\) in Newton's law of cooling. When the temperature of this object is \(0^{\circ} \mathrm{F}\), it is placed in a medium whose temperature is changing in time according to $$T_{m}(t)=80 e^{-t / 20}$$ (a) Using Newton's law of cooling, show that the temperature of the object at time \(t\) is $$T(t)=80\left(e^{-t / 40}-e^{-t / 20}\right)$$ (b) What happens to the temperature of the object as \(t \rightarrow+\infty ?\) Is this reasonable? (c) Determine the time, \(t_{\max },\) when the temperature of the object is a maximum. Find \(T\left(t_{\max }\right)\) and \(T_{m}\left(t_{\max }\right)\) (d) Make a sketch to depict the behavior of \(T(t)\) and \(T_{m}(t)\)

Determine a solution to the differential equation $$\left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}+4 y=0$$ of the form \(y(x)=a_{0}+a_{1} x+a_{2} x^{2}\) satisfying the normalization condition \(y(1)=1\).

At \(4 \mathrm{p} . \mathrm{m} .\) a hot coal was pulled out of a furnace and allowed to cool at room temperature \(\left(75^{\circ} \mathrm{F}\right) .\) If, after 10 minutes, the temperature of the coal was \(415^{\circ} \mathrm{F}\), and after 20 minutes, its temperature was \(347^{\circ} \mathrm{F}\), find the following: (a) The temperature of the furnace. (b) The time when the temperature of the coal was \(100^{\circ} \mathrm{F}.\)

Sketch the slope field and some representative solution curves for the given differential equation. $$y^{\prime}=x^{2} y$$

Find all solutions to $$x \frac{d y}{d x}-y=\sqrt{4 x^{2}-y^{2}}, \quad x>0$$

Determine an integrating factor for the given differential equation, and hence find the general solution. $$x y[2 \ln (x y)+1] d x+x^{2} d y=0, \quad x>0$$

The differential equation $$\frac{d T}{d t}=-k_{1}\left[T-T_{m}(t)\right]+A_{0}$$ where \(k_{1}\) and \(A_{0}\) are positive constants, can be used to model the temperature variation \(T(t)\) in a building. In this equation, the first term on the right-hand side gives the contribution due to the variation in the outside temperature, and the second term on the right-hand side gives the contribution due to the heating effect from internal sources such as machinery, lighting, people, etc. Consider the case when $$T_{m}(t)=A-B \cos \omega t, \quad \omega=\pi / 12$$ where \(A\) and \(B\) are constants, and \(t\) is measured in hours. (a) Make a sketch of \(T_{m}(t) .\) Taking \(t=0\) to correspond to midnight, describe the variation of the external temperature over a 24 -hour period. (b) With \(T_{m}\) given in \((1.6 .14),\) solve \((1.6 .13)\) subject to the initial condition \(T(0)=T_{0}\)

Show that the given relation defines an implicit solution to the given differential equation, where \(c\) is an arbitrary constant. $$x \sin y-e^{x}=c, \quad y^{\prime}=\frac{e^{x}-\sin y}{x \cos y}$$.

Sketch the slope field and some representative solution curves for the given differential equation. $$y^{\prime}=x^{2} \cos y$$

An object that is initially thrown vertically upward with a speed of 2 meters/second from a height of \(h\) meters takes 10 seconds to reach the ground. Set up and solve the initial-value problem that governs the motion of the object, and determine \(h\)

(a) Show that the general solution to the differential equation $$ \frac{d y}{d x}=\frac{x+a y}{a x-y} $$ can be written in polar form as \(r=k e^{a \theta}\) (b) For the particular case when \(a=1 / 2,\) determine the solution satisfying the initial condition \(y(1)=1,\) and find the maximum \(x\) -interval on which this solution is valid. (Hint: When does the solution curve have a vertical tangent?) (c) \(\diamond\) On the same set of axes, sketch the spiral corresponding to your solution in (b), and the line \(y=x / 2\). Thus verify the \(x\) -interval obtained in (b) with the graph.

Determine which of the five types of differential equations we have studied the given equation falls into (see Table \(1.12 .1),\) and use an appropriate technique to find the general solution. $$y[\ln (y / x)+1] d x-x d y=0$$

Determine an integrating factor for the given differential equation, and hence find the general solution. $$y d x-\left(2 x+y^{4}\right) d y=0$$

This problem demonstrates the variation-ofparameters method for first-order linear differential equations. Consider the first-order linear differential equation $$y^{\prime}+p(x) y=q(x)$$ (a) Show that the general solution to the associated homogeneous equation $$y^{\prime}+p(x) y=0$$ is $$y_{H}(x)=c_{1} e^{-\int p(x) d x}$$ (b) Determine the function \(u(x)\) such that $$y(x)=u(x) e^{-\int p(x) d x}$$ is a solution to \((1.6 .15),\) and hence derive the general solution to (1.6.15).

Show that the given relation defines an implicit solution to the given differential equation, where \(c\) is an arbitrary constant. $$x y^{2}+2 y-x=c, \quad y^{\prime}=\frac{1-y^{2}}{2(1+x y)}$$.

Sketch the slope field and some representative solution curves for the given differential equation. $$y^{\prime}=x^{2}+y^{2}$$

An object that is released from a height \(h\) meters above the ground with a vertical velocity of \(v_{0}\) meters/second hits the ground after \(t_{0}\) seconds. Neglecting frictional forces, set up and solve the initial-value problem governing the motion, and use your solution to show that $$ v_{0}=\frac{1}{2 t_{0}}\left(2 h-g t_{0}^{2}\right) $$

Determine which of the five types of differential equations we have studied the given equation falls into (see Table \(1.12 .1),\) and use an appropriate technique to find the general solution. $$\left(1+2 x e^{y}\right) d x-\left(e^{y}+x\right) d y=0$$

Determine the values of the constants \(r\) and \(s\) such that \(I(x, y)=x^{r} y^{s}\) is an integrating factor for the given differential equation. $$\left(y^{-1}-x^{-1}\right) d x+\left(x y^{-2}-2 y^{-1}\right) d y=0$$

Use the technique derived in the previous problem to solve the given differential equation. $$y^{\prime}+x^{-1} y=\cos x, \quad x > 0$$

Show that the given relation defines an implicit solution to the given differential equation, where \(c\) is an arbitrary constant. $$e^{x y}-x=c, \quad y^{\prime}=\frac{1-y e^{x y}}{w}$$. Determine the solution with \(y(1)=0\).

According to Newton's law of cooling (see Section 1.1), the temperature of an object at time \(t\) is governed by the differential equation $$ \frac{d T}{d t}=-k\left(T-T_{m}\right) $$ where \(T_{m}\) is the temperature of the surrounding medium, and \(k\) is a constant. Consider the case when \(T_{m}=70\) and \(k=1 / 80 .\) Sketch the corresponding slope field and some representative solution curves. What happens to the temperature of the object as \(t \rightarrow \infty .\) Note that this result is independent of the initial temperature of the object.

Determine which of the five types of differential equations we have studied the given equation falls into (see Table \(1.12 .1),\) and use an appropriate technique to find the general solution. $$y^{\prime}+y \sin x=\sin x$$

Determine the values of the constants \(r\) and \(s\) such that \(I(x, y)=x^{r} y^{s}\) is an integrating factor for the given differential equation. $$2 y\left(y+2 x^{2}\right) d x+x\left(4 y+3 x^{2}\right) d y=0$$

Use the technique derived in the previous problem to solve the given differential equation. $$y^{\prime}+y=e^{-2 x}$$

In a certain chemical reaction \(9 \mathrm{g}\) of \(\mathrm{C}\) are formed when \(6 \mathrm{g}\) of A combine with \(3 \mathrm{g}\) of \(\mathrm{B}\). Initially there are \(20 \mathrm{g}\) of both \(\mathrm{A}\) and \(\mathrm{B},\) and after \(10 \mathrm{min}, 15 \mathrm{g}\) of \(\mathrm{C}\) has been produced. Determine the amount of C that is produced in 20 min.

Show that the given relation defines an implicit solution to the given differential equation, where \(c\) is an arbitrary constant. $$e^{y / x}+x y^{2}-x=c, \quad y^{\prime}=\frac{x^{2}\left(1-y^{2}\right)+y e^{y / x}}{x\left(e^{y / x}+2 x^{2} y\right)}$$.

Determine the slope field and some representative solution curves for the given differential equation. $$\diamond y^{\prime}=-2 x y$$

Determine which of the five types of differential equations we have studied the given equation falls into (see Table \(1.12 .1),\) and use an appropriate technique to find the general solution. $$\left(3 y^{2}+x^{2}\right) d x-2 x y d y=0$$

Determine the values of the constants \(r\) and \(s\) such that \(I(x, y)=x^{r} y^{s}\) is an integrating factor for the given differential equation. $$y\left(5 x y^{2}+4\right) d x+x\left(x y^{2}-1\right) d y=0$$