Chapter 1

Solve the given differential equation. $$y^{\prime \prime}+y^{\prime} \tan x=\left(y^{\prime}\right)^{2}$$

Solve the given differential equation. $$\left(y^{2}-2 x\right) d x+2 x y d y=0$$

Solve the given differential equation. $$\frac{d y}{d x}=\frac{2 x(y-1)}{x^{2}+3}$$

A tank initially contains \(10 \mathrm{L}\) of a salt solution. Water flows into the tank at a rate of \(3 \mathrm{L} / \mathrm{min},\) and the well-stirred mixture flows out at a rate of 2 L/min. After 5 minutes, the concentration of salt in the tank is \(0.2 \mathrm{g} / \mathrm{L} .\) Find: (a) The amount of salt in the tank initially. (b) The volume of solution in the tank when the concentration of salt is \(0.1 \mathrm{g} / \mathrm{L}\).

In the logistic population model \((1.5 .3),\) if \(P\left(t_{1}\right)=P_{1}\) and \(P\left(2 t_{1}\right)=P_{2},\) then it can be shown (through some tedious algebra to derive by hand, although easy on a computer algebra system) that $$\begin{aligned}r &=\frac{1}{t_{1}} \ln \left[\frac{P_{2}\left(P_{1}-P_{0}\right)}{P_{0}\left(P_{2}-P_{1}\right)}\right] \\\c=& \frac{P_{1}\left[P_{1}\left(P_{0}+P_{2}\right)-2 P_{0} P_{2}\right]}{P_{1}^{2}-P_{0} P_{2}}\end{aligned}$$ These formulas will be used. An animal sanctuary had an initial population of 50 animals. After two years, the population was \(62,\) while after four years it was \(76 .\) Using the logistic population model, determine the carrying capacity and the number of animals in the sanctuary after 20 years.

Determine whether the differential equation is linear or nonlinear. $$\sqrt{x} y^{\prime \prime}+\frac{1}{y^{\prime}} \ln x=3 x^{3}$$.

Determine the differential equation giving the slope of the tangent line at the point \((x, y)\) for the given family of curves. $$x^{2}+y^{2}=2 c x$$

Verify that \(y(x)=x /(x+1)\) is a solution to the differential equation $$ y+\frac{d^{2} y}{d x^{2}}=\frac{d y}{d x}+\frac{x^{3}+2 x^{2}-3}{(1+x)^{3}} $$

Consider the family of curves $$ x^{2}+3 y^{2}=2 c y $$ (a) Show that the differential equation of this family is \(\frac{d y}{d x}=\frac{2 x y}{x^{2}-3 y^{2}}\) (b) Determine the orthogonal trajectories to the family (1.12.5).

Solve the given differential equation. $$\frac{d^{2} x}{d t^{2}}=\left(\frac{d x}{d t}\right)^{2}+2 \frac{d x}{d t}$$

Solve the given differential equation. $$\left(4 e^{2 x}+2 x y-y^{2}\right) d x+(x-y)^{2} d y=0$$

Solve the given differential equation. $$\begin{aligned} &2\left(\cos ^{2} x\right) y^{\prime}+y \sin 2 x=4 \cos ^{4} x\\\ &0 \leq x<\pi / 2 \end{aligned}$$

Determine whether the given function is homogeneous of degree zero. Rewrite those that are as functions of the single variable \(V=y / x\). $$f(x, y)=\frac{\sqrt{x^{2}+y^{2}}}{x}, \quad x<0$$

A tank initially contains \(w\) liters of a solution in which is dissolved \(A_{0}\) grams of chemical. A solution containing \(k\) g/L of this chemical flows into the tank at a rate of \(r\) L/min, and the mixture flows out at the same rate. (a) Show that the amount of chemical, \(A(t),\) in the tank at time \(t\) is $$ A(t)=e^{-(r t) / w}\left[k w\left(e^{(r t) / w}-1\right)+A_{0}\right] $$ (b) Show that as \(t \rightarrow \infty,\) the concentration of chemical in the tank approaches \(k\) g/L. Is this result reasonable? Explain.

In the logistic population model \((1.5 .3),\) if \(P\left(t_{1}\right)=P_{1}\) and \(P\left(2 t_{1}\right)=P_{2},\) then it can be shown (through some tedious algebra to derive by hand, although easy on a computer algebra system) that $$\begin{aligned}r &=\frac{1}{t_{1}} \ln \left[\frac{P_{2}\left(P_{1}-P_{0}\right)}{P_{0}\left(P_{2}-P_{1}\right)}\right] \\\c=& \frac{P_{1}\left[P_{1}\left(P_{0}+P_{2}\right)-2 P_{0} P_{2}\right]}{P_{1}^{2}-P_{0} P_{2}}\end{aligned}$$ These formulas will be used. (a) Using Equations \((1.5 .5)\) and \((1.5 .6),\) and the fact that \(r\) and \(C\) are positive, derive two incqualities that \(P_{0}, P_{1}, P_{2}\) must satisfy in order for there to be a solution to the logistic equation satisfying the conditions $$P(0)=P_{0}, \quad P\left(t_{1}\right)=P_{1}, \quad P\left(2 t_{1}\right)=P_{2}$$ (b) The initial population in a town is \(10,000 .\) After five years, this has grown to be \(12,000,\) while after ten years, the population is \(18,000 .\) Is there a solution to the logistic equation that fits this data?

Verify that the given function is a solution to the given differential equation \(\left(c_{1} \text { and } c_{2}\right.\) are arbitrary constants), and state the maximum interval over which the solution is valid. $$y(x)=c_{1} e^{-5 x}+c_{2} e^{5 x}, \quad y^{\prime \prime}-25 y=0$$.

Solve the given differential equation. $$y-x \frac{d y}{d x}=3-2 x^{2} \frac{d y}{d x}$$

Determine the differential equation giving the slope of the tangent line at the point \((x, y)\) for the given family of curves. $$(x-c)^{2}+(y-c)^{2}=2 c^{2}$$

Verify that \(y(x)=e^{x} \sin x\) is a solution to the differential equation $$ 2 y \cot x-\frac{d^{2} y}{d x^{2}}=0 $$

Sketch the slope field and some representative solution curves for the given differential equation. $$y^{\prime}=y(y-1)^{2}$$

Solve the given differential equation. $$y^{\prime \prime}-2 x^{-1} y^{\prime}=6 x^{4}$$

Solve the given differential equation. $$\left(\frac{1}{x}-\frac{y}{x^{2}+y^{2}}\right) d x+\frac{x}{x^{2}+y^{2}} d y=0$$

Determine whether the given function is homogeneous of degree zero. Rewrite those that are as functions of the single variable \(V=y / x\). $$f(x, y)=\frac{\sqrt{x^{2}+4 y^{2}}-x+y}{x+3 y}, \quad x, y \neq 0$$

Solve the given differential equation. $$y^{\prime}+\frac{1}{x \ln x} y=9 x^{2}$$

Of the 1500 passengers, crew, and staff that board a cruise ship, 5 have the flu. After one day of sailing, the number of infected people has risen to \(10 .\) Assuming that the rate at which the flu virus spreads is proportional to the product of the number of infected individuals and the number not yet infected, determine how many people will have the flu at the end of the 14-day cruise. Would you like to be a member of the customer relations department for the cruise line the day after the ship docks?

Verify that the given function is a solution to the given differential equation \(\left(c_{1} \text { and } c_{2}\right.\) are arbitrary constants), and state the maximum interval over which the solution is valid. $$y(x)=c_{1} \cos 2 x+c_{2} \sin 2 x, \quad y^{\prime \prime}+4 y=0$$.

Solve the given differential equation. $$\frac{d y}{d x}=\frac{\cos (x-y)}{\sin x \sin y}-1$$

Determine the differential equation giving the slope of the tangent line at the point \((x, y)\) for the given family of curves. $$2 c y=x^{2}-c^{2}$$

By writing Equation (1.1.7) in the form $$ \frac{1}{T-T_{m}} \frac{d T}{d t}=-k $$ and using \(u^{-1} \frac{d u}{d t}=\frac{d}{d t}(\ln u),\) derive \((1.1 .8)\)

Sketch the slope field and some representative solution curves for the given differential equation. $$y^{\prime}=(y-3)(y+1)$$

Solve the given differential equation. $$t \frac{d^{2} x}{d t^{2}}=2\left(t+\frac{d x}{d t}\right)$$

Solve the given differential equation. $$\left(y^{2}-2 x\right) d x+2 x y d y=0$$

Solve the given differential equation. $$\frac{d y}{d x}=\frac{x\left(y^{2}-1\right)}{2(x-2)(x-1)}$$

Solve the given differential equation. $$y^{\prime}-y \tan x=8 \sin ^{3} x$$

Consider the population model $$\frac{d P}{d t}=r(P-T) P, \quad P(0)=P_{0}$$ where \(r, T,\) and \(P_{0}\) are positive constants. (a) Perform a qualitative analysis of the differential equation in the initial-value problem (1.5.7) following the steps used in the text for the logistic equation. Identify the equilibrium solutions, the isoclines, and the behavior of the slope and concavity of the solution curves. (b) Using the information obtained in (a), sketch the slope field for the differential equation and include representative solution curves. (c) What predictions can you make regarding the behavior of the population? Consider the cases \(P_{0}T .\) The constant \(T\) is called the threshold level. Based on your predictions, why is this an appropriate term to use for \(T ?\)

Verify that the given function (or relation) defines a solution to the given differential equation and sketch some of the solution curves. If an initial condition is given, label the solution curve corresponding to the resulting unique solution. (In these problems, \(c\) denotes an arbitrary constant.) $$x^{2}+y^{2}=c, \quad y^{\prime}=-x / y$$

A glass of water whose temperature is \(50^{\circ} \mathrm{F}\) is taken outside at noon on a day whose temperature is constant at \(70^{\circ} \mathrm{F}\). If the water's temperature is \(55^{\circ} \mathrm{F}\) at \(2 \mathrm{p} . \mathrm{m} .,\) do you expect the water's temperature to reach \(60^{\circ} \mathrm{F}\) before \(4 \mathrm{p} . \mathrm{m}\). or after \(4 \mathrm{p} . \mathrm{m} . ?\) Use Newton's law of cooling to explain your answer.

Sketch the slope field and some representative solution curves for the given differential equation. $$y^{\prime}=y(2-y)(1-y)$$

Solve the given differential equation. \(y^{\prime \prime}-\alpha\left(y^{\prime}\right)^{2}-\beta y^{\prime}=0,\) where \(\alpha\) and \(\beta\) are nonzero constants.

Solve the given differential equation. $$\left(2 y^{2} e^{2 x}+3 x^{2}\right) d x+2 y e^{2 x} d y=0$$

Solve the given differential equation. $$(3 x-2 y) \frac{d y}{d x}=3 y$$

Solve the given differential equation. $$t \frac{d x}{d t}+2 x=4 e^{t}, \quad t>0$$

Consider the RC circuit which has \(R=5 \Omega, C=\frac{1}{50}\) \(\mathrm{F},\) and \(E(t)=100 \mathrm{V} .\) If the capacitor is uncharged initially, determine the current in the circuit for \(t \geq 0\).

Solve the given differential equation. $$\frac{d y}{d x}=\frac{x^{2} y-32}{16-x^{2}}+2$$

Verify that the given function (or relation) defines a solution to the given differential equation and sketch some of the solution curves. If an initial condition is given, label the solution curve corresponding to the resulting unique solution. (In these problems, \(c\) denotes an arbitrary constant.) $$y=c x^{3}, \quad y^{\prime}=3 y / x, \quad y(2)=8$$

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

As a modification to the population model considered in the previous two problems, suppose that \(P(t)\) satisfies the initial-value problem $$\frac{d P}{d t}=r(C-P)(P-T) P, \quad P(0)=P_{0}$$ where \(r, C, T, P_{0}\) are positive constants, and \(0 < T< \) C. Perform a qualitative analysis of this model. Sketch the slope ficld, and some representative solution curves in the three cases \(0 < P_{0} < T, T < P_{0} < C,\) and \(P_{0} > C .\) Describe the behavior of the corresponding solutions. The next two problems consider the Gompertz population model, which is governed by the initial-value problem $$\frac{d P}{d t}=r P(\ln C-\ln P), \quad P(0)=P_{0}$$ where \(r, C,\) and \(P_{0}\) are positive constants.

Solve the given differential equation. $$y^{\prime \prime}-2 x^{-1} y^{\prime}=18 x^{4}$$

Solve the given differential equation. $$\left(y^{2}+\cos x\right) d x+(2 x y+\sin y) d y=0$$

Solve the given differential equation. $$y^{\prime}=\frac{(x+y)^{2}}{2 x^{2}}$$