Chapter 2

Consider the autonomous DE \(d y / d x=(2 / \pi) y-\sin y\). Determine the critical points of the equation. Discuss a way of obtaining a phase portrait of the equation. Classify the critical points as asymptotically stable, unstable, or semi-stable.

$$ \begin{aligned} &\left(1+t^{2}\right) \frac{d x}{d t}+x=\tan ^{-1} t, \quad x(0)=4\\\ &\text { [Hint: In your solution let } \left.u=\tan ^{-1} t .\right] \end{aligned} $$

Solve the given differential equation by finding, as in Example 4 , an appropriate integrating factor. $$ y(x+y+1) d x+(x+2 y) d y=0 $$

Find a solution of \(x \frac{d y}{d x}=y^{2}-y\) that passes through the indicated points. (a) \((0,1)\) (b) \((0,0)\) (c) \(\left(\frac{1}{2}, \frac{1}{2}\right)\) (d) \(\left(2, \frac{1}{4}\right)\)

A critical point \(c\) of an autonomous first-order \(\mathrm{DE}\) is said to be isolated if there exists some open interval that contains \(c\) but no other critical point. Discuss: Can there exist an autonomous DE of the form given in (1) for which every critical point is nonisolated? Do not think profound thoughts.

A 200-volt electromotive force is applied to an \(R C\)-series circuit in which the resistance is 1000 ohms and the capacitance is \(5 \times 10^{-6}\) farad. Find the charge \(q(t)\) on the capacitor if \(i(0)=0.4\). Determine the charge and current at \(t=0.005 \mathrm{~s}\). Determine the charge as \(t \rightarrow \infty\).

Find a solution of \(x \frac{a y}{d x}=y^{2}-y\) that passes through the indicated points. (a) \((0,1)\) (b) \((0,0)\) (c) \(\left(\frac{1}{2}, \frac{1}{2}\right)\) (d) \(\left(2, \begin{array}{l}1 \\ 4\end{array}\right)\)

Solve the given initial-value problem. Give the largest interval \(\boldsymbol{l}\) over which the solution is defined. $$ \left(1+t^{2}\right) \frac{d x}{d t}+x=\tan ^{-1} t, \quad x(0)=4 $$

A critical point \(c\) of an autonomous first-order DE is said to be isolated if there exists some open interval that contains \(c\) but no other critical point. Discuss: Can there exist an autonomous DE of the form given in (1) for which every critical point is nonisolated? Do not think profound thoughts.

An electromotive force $$ E(t)=\left\\{\begin{array}{ll} 120, & 0 \leq t \leq 20 \\ 0, & t>20 \end{array}\right. $$ is applied to an \(L R\) -series circuit in which the inductance is 20 henries and the resistance is 2 ohms. Find the current \(i(t)\) if \(i(0)=0\).

Solve the given differential equation by finding, as in Example 4 , an appropriate integrating factor. $$ 6 x y d x+\left(4 y+9 x^{2}\right) d y=0 $$

An electromotive force $$ E(t)= \begin{cases}120, & 0 \leq t \leq 20 \\ 0, & t>20\end{cases} $$ is applied to an \(L R\)-series circuit in which the inductance is 20 henries and the resistance is \(2 \mathrm{ohms}\). Find the current \(i(t)\) if \(i(0)=0\).

Suppose an \(R C\) -series circuit has a variable resistor. If the resistance at time \(t\) is given by \(R=k_{1}+k_{2} t\), where \(k_{1}\) and \(k_{2}\) are known positive constants, then (9) becomes $$ \left(k_{1}+k_{2} t\right) \frac{d q}{d t}+\frac{1}{C} q=E(t) $$ If \(E(t)=E_{0}\) and \(q(0)=q_{0}\), where \(E_{0}\) and \(q_{0}\) are constants, show that $$ q(t)=E_{0} C+\left(q_{0}-E_{0} C\right)\left(\frac{k_{1}}{k_{1}+k_{2} t}\right)^{1 / C k_{2}} $$

Solve the given differential equation by finding, as in Example 4 , an appropriate integrating factor. $$ \cos x d x+\left(1+\frac{2}{y}\right) \sin x d y=0 $$

Show that an implicit solution of $$ 2 x \sin ^{2} y d x-\left(x^{2}+10\right) \cos y d y=0 $$ is given by \(\ln \left(x^{2}+10\right) \csc y=c\). Find the constant solutions, if any, that were lost in the solution of the differential equation.

The differential equation $$ \frac{d y}{d x}=P(x)+Q(x) y+R(x) y^{2} $$ is known as Riccati's equation. (a) A Riccati equation can be solved by a succession of two substitutions provided we know a particular solution \(y_{1}\) of the equation. Show that the substitution \(y=y_{1}+u\) reduces Riccati's equation to a Bernoulli equation (4) with \(n=2\). The Bernoulli equation can then be reduced to a linear equation by the substitution \(w=u^{-1}\).

Often a radical change in the form of the solution of a differential equation corresponds to a very small change in either the initial condition or the equation itself. In Problems, find an explicit solution of the given initial- value problem. Use a graphing utility to plot the graph of each solution. Compare each solution curve in a neighborhood of \((0,1)\). \frac{d y}{d x}=(y-1)^{2}, \quad y(0)=1

Solve the given differential equation by finding, as in Example 4, an appropriate integrating factor. $$ \left(10-6 y+e^{-3 x}\right) d x-2 d y=0 $$

Suppose a small cannonball weighing \(16 \mathrm{lb}\) is shot vertically upward with an initial velocity \(v_{0}=300 \mathrm{ft} / \mathrm{s}\). The answer to the question, "How high does the cannonball go?" depends on whether we take air resistance into account. (a) Suppose air resistance is ignored. If the positive direction is upward, then a model for the state of the cannonball is given by \(d^{2} s / d t^{2}=-g\) (equation (12) of Section 1.3). Since \(d s / d t=v(t)\) the last differential equation is the same as \(d v / d t=-g\), where we take \(g=32 \mathrm{ft} / \mathrm{s}^{2} .\) Find the velocity \(v(t)\) of the cannonball at time \(t\). (b) Use the result obtained in part (a) to determine the height \(s(t)\) of the cannonball measured from ground level. Find the maximum height attained by the cannonball.

Devise an appropriate substitution to solve $$ x y^{\prime}=y \ln (x y) $$

Solve the given differential equation by finding, as in Example 4 , an appropriate integrating factor. $$ \left(y^{2}+x y^{3}\right) d x+\left(5 y^{2}-x y+y^{3} \sin y\right) d y=0 $$

Often a radical change in the form of the solution of a differential equation corresponds to a very small change in either the initial condition or the equation itself. In Problems, find an explicit solution of the given initial- value problem. Use a graphing utility to plot the graph of each solution. Compare each solution curve in a neighborhood of \((0,1)\). $$ \frac{d y}{d x}=(y-1)^{2}, \quad y(0)=1.01 $$

How High? — No Air Resistance Suppose a small cannonball weighing \(16 \mathrm{lb}\) is shot vertically upward with an initial velocity \(v_{0}=300 \mathrm{ft} / \mathrm{s}\). The answer to the question, "How high does the cannonball go?" depends on whether we take air resistance into account. (a) Suppose air resistance is ignored. If the positive direction is upward, then a model for the state of the cannonball is given by \(d^{2} s / d t^{2}=-g\) (equation (12) of Section 1.3). Since \(d s / d t=v(t)\) the last differential equation is the same as \(d v / d t=-g\), where we take \(g=32 \mathrm{ft} / \mathrm{s}^{2}\). Find the velocity \(v(t)\) of the cannonball at time \(t\). (b) Use the result obtained in part (a) to determine the height \(s(t)\) of the cannonball measured from ground level. Find the maximum height attained by the cannonball.

In the study of population dynamics one of the most famous models for a growing but bounded population is the logistic equation $$ \frac{d P}{d t}=P(a-b P) $$ where \(a\) and \(b\) are positive constants. Although we will come back to this equation and solve it by an alternative method in Section \(2.8\), solve the \(\mathrm{DE}\) this first time using the fact that it is a Bernoulli equation.

Solve the given initial-value problem by finding, as in Example 4, an appropriate integrating factor. $$ x d x+\left(x^{2} y+4 y\right) d y=0, \quad y(4)=0 $$

Often a radical change in the form of the solution of a differential equation corresponds to a very small change in either the initial condition or the equation itself. In Problems, find an explicit solution of the given initial- value problem. Use a graphing utility to plot the graph of each solution. Compare each solution curve in a neighborhood of \((0,1)\). $$ \frac{d y}{d x}=(y-1)^{2}+0.01, \quad y(0)=1 $$

Population Growth In the study of population dynamics one of the most famous models for a growing but bounded population is the logistic equation $$ \frac{d P}{d t}=P(a-b P), $$ where \(a\) and \(b\) are positive constants. Although we will come back to this equation and solve it by an alternative method in Section 2.8, solve the DE this first time using the fact that it is a Bernoulli equation.

Consider the initial-value problem \(y^{\prime}+e^{x} y=f(x), y(0)=1\) Express the solution of the IVP for \(x>0\) as a nonelementary integral when \(f(x)=1\). What is the solution when \(f(x)=0 ?\) When \(f(x)=e^{x} ?\)

Solve the given initial-value problem by finding, as in Example 4, an appropriate integrating factor. $$ \left(x^{2}+y^{2}-5\right) d x=(y+x y) d y, \quad y(0)=1 $$

Often a radical change in the form of the solution of a differential equation corresponds to a very small change in either the initial condition or the equation itself. In Problems, find an explicit solution of the given initial- value problem. Use a graphing utility to plot the graph of each solution. Compare each solution curve in a neighborhood of \((0,1)\). $$ \frac{d y}{d x}=(y-1)^{2}-0.01, \quad y(0)=1 $$

The differential equation in Example 3 is a well-known population model. Suppose the DE is changed to $$ \frac{d P}{d t}=P(a P-b) $$ where \(a\) and \(b\) are positive constants. Discuss what happens to the population \(P\) as time \(t\) increases.

Population Model The differential equation in Example 3 is a well-known population model. Suppose the DE is changed to $$ \frac{d P}{d t}=P(a P-b) $$ where \(a\) and \(b\) are positive constants. Discuss what happens to the population \(P\) as time \(t\) increases.

Express the solution of the initial-value problem \(y^{\prime}-2 x y=1\), \(y(1)=1\), in terms of \(\operatorname{erf}(x)\)

(a) Show that a one-parameter family of solutions of the equation $$ \left(4 x y+3 x^{2}\right) d x+\left(2 y+2 x^{2}\right) d y=0 $$ is \(x^{3}+2 x^{2} y+y^{2}=c\). (b) Show that the initial conditions \(y(0)=-2\) and \(y(1)=1\) determine the same implicit solution. (c) Find explicit solutions \(y_{1}(x)\) and \(y_{2}(x)\) of the differential equation in part (a) such that \(y_{1}(0)=-2\) and \(y_{2}(1)=1\). Use a graphing utility to graph \(y_{1}(x)\) and \(y_{2}(x)\).

Every autonomous first-order equation \(d y / d x=f(y)\) is separable. Find explicit solutions \(y_{1}(x), y_{2}(x), y_{3}(x)\), and \(y_{4}(x)\) of the differential equation \(d y / d x=y-y^{3}\) that satisfy, in turn, the initial conditions \(y_{1}(0)=2, y_{2}(0)=\frac{1}{2}, y_{3}(0)=-\frac{1}{2}\), and \(y_{4}(0)=-2\). Use a graphing utility to plot the graphs of each solution. Compare these graphs with those predicted in Problem 19 of Exercises \(2.1\). Give the exact interval of definition for each solution.

The differential equation \(d P / d t=\) \((k \cos t) P\), where \(k\) is a positive constant, is a mathematical model for a population \(P(t)\) that undergoes yearly seasonal fluctuations. Solve the equation subject to \(P(0)=P_{0}\). Use a graphing utility to obtain the graph of the solution for different choices of \(P_{0}\)

When all the curves in a family \(G\left(x, y, c_{1}\right)=0\) intereect orthogonally, all the curves in another family \(H\left(x, y, c_{2}\right)=0\), the families are said to be orthogonal trajectories of each other. See FICURE 2.R.11. If \(d y / d x=f(x, y)\) is the differential equation of one family, then the differential equation for the orthogonal trajectores of this family is \(d y / d x=-1 / f(x, y)\). In Problems 39 and 40 , find the differential equation of the given family. Find the orthogonal trajectories of this family. Use a graphing utility to graph both families on the same set of coordinate axes. FGURE 2.I.11 Orthogonal trajectories $$ y=\frac{1}{x+c_{1}} $$

Fluctuating Population The differential equation \(d P / d t=\) \((k \cos t) P\), where \(k\) is a positive constant, is a mathematical model for a population \(P(t)\) that undergoes yearly seasonal fluctuations. Solve the equation subject to \(P(0)=P_{0}\). Use a graphing utility to obtain the graph of the solution for different choices of \(P_{0}\).

Consider the concept of anintegrating factor used in Problems 29-38. Are the two equations \(M d x+N d y=0\) and \(\mu M d x+\) \(\mu N d y=0\) necessarily equivalent in the sense that a solution of one is also a solution of the other? Discuss.

In one model of the changing population \(P(t)\) of a community, it is assumed that $$ \frac{d P}{d t}=\frac{d B}{d t}-\frac{d D}{d t} $$ where \(d B / d t\) and \(d D / d t\) are the birth and death rates, respectively. (a) Solve for \(P(t)\) if \(d B / d t=k_{1} P\) and \(d D / d t=k_{2} P\). (b) Analyze the cases \(k_{1}>k_{2}, k_{1}=k_{2}\), and \(k_{1}

(a) Find an explicit solution of the initial-value problem $$ \frac{d y}{d x}=\frac{2 x+1}{2 y}, \quad y(-2)=-1 $$ (b) Use a graphing utility to plot the graph of the solution in part (a). Use the graph to estimate the interval \(I\) of definition of the solution. (c) Determine the exact interval \(I\) of definition by analytical methods.

Population Model In one model of the changing population \(P(t)\) of a community, it is assumed that $$ \frac{d P}{d t}=\frac{d B}{d t}-\frac{d D}{d t} $$ where \(d B / d t\) and \(d D / d t\) are the birth and death rates, respectively. (a) Solve for \(P(t)\) if \(d B / d t=k_{1} P\) and \(d D / d t=k_{2} P\). (b) Analyze the cases \(k_{1}>k_{2}, k_{1}=k_{2}\), and \(k_{1}

When forgetfulness is taken into account, the rate of memorization of a subject is given by $$ \frac{d A}{d t}=k_{1}(M-A)-k_{2} A $$ where \(k_{1}>0, k_{2}>0, A(t)\) is the amount to be memorized in time \(t, M\) is the total amount to be memorized, and \(M-A\) is the amount remaining to be memorized. See Problems 25 and 26 in Exercises \(1.3\). (a) Since the DE is autonomous, use the phase portrait concept of Section \(2.1\) to find the limiting value of \(A(t)\) as \(t \rightarrow \infty\). Interpret the result. (b) Solve for \(A(t)\) subject to \(A(0)=0\). Sketch the graph of \(A(t)\) and verify your prediction in part (a).

Memorization When forgetfulness is taken into account, the rate of memorization of a subject is given by $$ \frac{d A}{d t}=k_{1}(M-A)-k_{2} A $$ where \(k_{1}>0, k_{2}>0, A(t)\) is the amount to be memorized in time \(t, M\) is the total amount to be memorized, and \(M-A\) is the amount remaining to be memorized. See Problems 25 and 26 in Exercises 1.3. (a) Since the DE is autonomous, use the phase portrait concept of Section \(2.1\) to find the limiting value of \(A(t)\) as \(t \rightarrow \infty\). Interpret the result. (b) Solve for \(A(t)\) subject to \(A(0)=0\). Sketch the graph of \(A(t)\) and verify your prediction in part (a).

A mathematical model for the rate at which a drug disseminates into the bloodstream is given by \(d x / d t=r-k x\), where \(r\) and \(k\) are positive constants. The function \(x(t)\) describes the concentration of the drug in the bloodstream at time \(t\). (a) Since the \(\mathrm{DE}\) is autonomous, use the phase portrait concept of Section \(2.1\) to find the limiting value of \(x(t)\) as \(t \rightarrow \infty\) (b) Solve the \(\mathrm{DE}\) subject to \(x(0)=0\). Sketch the graph of \(x(t)\) and verify your prediction in part (a). At what time is the concentration one- half this limiting value?

Rocket Motion Suppose a small single-stage rocket of total mass \(m(t)\) is launched vertically and that the rocket consumes its fuel at a constant rate. If the positive direction is upward and if we take air resistance to be linear, then a differential equation for its velocity \(v(t)\) is given by $$ \frac{d v}{d t}+\frac{k-\lambda}{m_{0}-\lambda t} v=-g+\frac{R}{m_{0}-\lambda t^{\prime}} $$ where \(k\) is the drag coefficient, \(\lambda\) is the rate at which fuel is consumed, \(R\) is the thrust of the rocket, \(m_{0}\) is the total mass of the rocket at \(t=0\), and \(g\) is the acceleration due to gravity. See Problem 21 in Exercises \(1.3\). (a) Find the velocity \(v(t)\) of the rocket if \(m_{0}=200 \mathrm{~kg}\), \(R=2000 \mathrm{~N}, \lambda=1 \mathrm{~kg} / \mathrm{s}, g=9.8 \mathrm{~m} / \mathrm{s}^{2}, k=3 \mathrm{~kg} / \mathrm{s}\), and \(v(0)=0\) (b) Use \(d s / d t=v\) and the result in part (a) to find the height \(s(t)\) of the rocket at time \(t\).

(a) If \(a>0\), discuss the differences, if any, between the solutions of the initial-value problems consisting of the differential equation \(d y / d x=x / y\) and each of the initial conditions \(y(a)=a, y(a)=-a, y(-a)=a\), and \(y(-a)=-a .\) (b) Does the initial-value problem \(d y / d x=x / y, y(0)=0\) have a solution? (c) Solve \(d y / d x=x l y, y(1)=2\), and give the exact interval \(I\) of definition of its solution.

True or False: Every separable first-order equation \(d y / d x=g(x) h(y)\) is exact.

(a) The solution of the differential equation $$ \frac{2 x y}{\left(x^{2}+y^{2}\right)^{2}} d x+\left[1+\frac{y^{2}-x^{2}}{\left(x^{2}+y^{2}\right)^{2}}\right] d y=0 $$ is a family of curves that can be interpreted as streamlines of a fluid flow around a circular object whose boundary is described by the equation \(x^{2}+y^{2}=1\). Solve this \(\mathrm{DE}\) and note the solution \(f(x, y)=c\) for \(c=0\). (b) Use a CAS to plot the streamlines for \(c=0, \pm 0.2, \pm 0.4\) \(\pm 0.6\), and \(\pm 0.8\) in three different ways. First, use the contourplot of a CAS. Second, solve for \(x\) in terms of the variable \(y .\) Plot the resulting two functions of \(y\) for the given values of \(c\), and then combine the graphs. Third, use the CAS to solve a cubic equation for \(y\) in terms of \(x\).

(a) Construct a linear first-order differential equation of the form \(x y^{\prime}+a_{0}(x) y=g(x)\) for which \(y_{c}=c / x^{3}\) and \(y_{p}=x^{3}\). Give an interval on which \(y=x^{3}+c / x^{3}\) is the general solution of the DE. (b) Give an initial condition \(y\left(x_{0}\right)=y_{0}\) for the DE found in part (a) so that the solution of the IVP is \(y=x^{3}-1 / x^{3}\). Repeat if the solution is \(y=x^{3}+2 / x^{3}\). Give an interval \(I\) of definition of each of these solutions. Graph the solution curves. Is there an initial-value problem whose solution is defined on the interval \((-\infty, \infty)\) ? (c) Is each IVP found in part (b) unique? That is, can there be more than one IVP for which, say, \(y=x^{3}-1 / x^{3}, x\) in some interval \(I\), is the solution?