Chapter 2

Use a numerical solver to obtain a numerical solution curve for the given initial-value problem. First use Euler's method and then the RK4 method. Use \(h=0.25\) in each case. Superimpose both solution curves on the same coordinate axes. If possible, use a different color for each curve. Repeat, using \(h=0.1\) and \(h=0.05\). $$ y^{\prime}=y(10-2 y), \quad y(0)=1 $$

Consider the competition model defined by $$ \begin{aligned} &\frac{d x}{d t}=x(2-0.4 x-0.3 y) \\ &\frac{d y}{d t}=y(1-0.1 y-0.3 x) \end{aligned} $$ where the populations \(x(t)\) and \(y(t)\) are measured in the thousands and \(t\) in years. Use a numerical solver to analyze the populations over a long period of time for each of the cases: (a) \(x(0)=1.5, \quad y(0)=3.5\) (b) \(x(0)=1, \quad y(0)=1\) (c) \(x(0)=2\) \(y(0)=7\) (d) \(x(0)=4.5\), \(y(0)=0.5\)

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution. $$ (1+x) \frac{d y}{d x}-x y=x+x^{2} $$

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

Determine whether the given differential equation is exact. If it is exact, solve it. $$ \left(3 x^{2} y+e^{y}\right) d x+\left(x^{3}+x e^{y}-2 y\right) d y=0 $$

Use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points. \(\frac{d y}{d x}=1-\frac{y}{x}\) (a) \(y\left(-\frac{1}{2}\right)=2\) (b) \(y\left(\frac{3}{2}\right)=0\)

In Problems 9-16, solve the given differential equation. $$ \frac{d x}{d y}=-\frac{4 y^{2}+6 x y}{3 y^{2}+2 x} $$

$$ \text { In Problems 11-14, solve the given initial-value problem. } $$ $$ \left(x^{2}+2 y^{2}\right) \frac{d x}{d y}=x y, y(-1)=1 $$

In Problems 1-22, solve the given differential equation by separation of variables. $$ \sin 3 x d x+2 y \cos ^{3} 3 x d y=0 $$

Use a numerical solver and Euler'smethodto approximate \(y(1.0)\) where \(y(x)\) is the solution to \(y^{\prime}=2 x y^{2}, y(0)=1\). First use \(h=0.1\) and then \(h=0.05\). Repeat using the \(\mathrm{RK} 4\) method. Discuss what might cause the approximations of \(y(1.0)\) to differ so greatly.

Comsider the competition model defined by $$ \begin{aligned} &\frac{d x}{d t}=x(1-0.1 x-0.05 y) \\ &\frac{d y}{d t}=y(1.7-0.1 y-0.15 x) \end{aligned} $$ where the populations \(x(t)\) and \(y(t)\) are measured in the thousands and \(t\) in years. Use a numerical solver to analyze the populations over a long period of time for each of the cases: (a) \(x(0)=1\), \(y(0)=1\) (b) \(x(0)-4, \quad y(0)=10\) (c) \(x(0)=9\), \(y(0)=4\) (d) \(x(0)=5.5\), \(y(0)=3.5\)

A tank in the form of a right circular cylinder standing on end is leaking water through a circular hole in its bottom. As we saw in (10) of Section 1.3, when friction and contraction of water at the hole are ignored, the height \(h\) of water in the tank is described by $$ \frac{d h}{d t}=-\frac{A_{h}}{A_{w}} \sqrt{2 g h}, $$ where \(A_{w}\) and \(A_{h}\) are the cross-sectional areas of the water and the hole, respectively. (a) Solve for \(h(t)\) if the initial height of the water is \(H\). By hand, sketch the graph of \(h(t)\) and give its interval \(I\) of definition in terms of the symbols \(A_{w}, A_{h}\), and \(H\). Use \(g=32 \mathrm{ft} / \mathrm{s}^{2}\) (b) Suppose the tank is \(10 \mathrm{ft}\) high and has radius \(2 \mathrm{ft}\) and the circular hole has radius \(\frac{1}{2}\) in. If the tank is initially full, how long will it take to empty?

Solve the given initial-value problem. $$ \left(x+y e^{y / x}\right) d x-x e^{\mathrm{y} / x} d y=0, \quad y(1)=0 $$

Determine whether the given differential equation is exact. If it is exact, solve it. $$ x \frac{d y}{d x}=2 x e^{x}-y+6 x^{2} $$

In Problems 9-16, solve the given differential equation. $$ t \frac{d Q}{d t}+Q=t^{4} \ln t $$

Leaking Cylindrical Tank A tank in the form of a rightcircular cylinder standing on end is leaking water through a circular hole in its bottom. As we saw in (10) of Section 1.3, when friction and contraction of water at the hole are ignored, the height \(h\) of water in the tank is described by $$ \frac{d h}{d t}=-\frac{A_{h}}{A_{w}} \sqrt{2 g h} $$ where \(A_{w}\) and \(A_{h}\) are the cross-sectional areas of the water and the hole, respectively. (a) Solve for \(h(t)\) if the initial height of the water is \(H\). By hand, sketch the graph of \(h(t)\) and give its interval \(I\) of definition in terms of the symbols \(A_{w}, A_{h}\), and \(H\). Use \(g=32 \mathrm{ft} / \mathrm{s}^{2}\). (b) Suppose the tank is \(10 \mathrm{ft}\) high and has radius \(2 \mathrm{ft}\) and the circular hole has radius \(\frac{1}{2}\) in. If the tank is initially full, how long will it take to empty?

$$ \text { In Problems 11-14, solve the given initial-value problem. } $$ $$ \left(x+y e^{y / x}\right) d x-x e^{y / x} d y=0, \quad y(1)=0 $$

In Problems 1-22, solve the given differential equation by separation of variables. $$ \left(e^{y}+1\right)^{2} e^{-y} d x+\left(e^{x}+1\right)^{3} e^{-x} d y=0 $$

A thetmometer is talen from an inside room to the outside, where the air temperahre is \(5^{\circ} \mathrm{F}\). After 1 minute the thermometer reads \(55^{\circ} \mathrm{F}\), and afler 5 minutes it reads \(30^{\circ} \mathrm{F}\). What is the initial temperahire of the inside room?

Solve the given initial-value problem. $$ y d x+x(\ln x-\ln y-1) d y=0, \quad y(1)=e $$

Determine whether the given differential equation is exact. If it is exact, solve it. $$ \left(1-\frac{3}{y}+x\right) \frac{d y}{d x}+y=\frac{3}{x}-1 $$

\(x\left(1+y^{2}\right)^{1 / 2} d x=y\left(1+x^{2}\right)^{1 / 2} d y\)

In Problems 9-16, solve the given differential equation. $$ (2 x+y+1) y^{\prime}=1 $$

$$ \text { In Problems 11-14, solve the given initial-value problem. } $$ $$ y d x+x(\ln x-\ln y-1) d y=0, \quad y(1)=e $$

In Problems 1-22, solve the given differential equation by separation of variables. $$ x\left(1+y^{2}\right)^{1 / 2} d x=y\left(1+x^{2}\right)^{1 / 2} d y $$

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution. $$ x y^{\prime}+(1+x) y=e^{-x} \sin 2 x $$

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution. $$ y d x-4\left(x+y^{6}\right) d y=0 $$

Solve the given differential equation by using an appropriate substitution. $$ x \frac{d y}{d x}+y=\frac{1}{y^{2}} $$

Determine whether the given differential equation is exact. If it is exact, solve it. $$ \left(x^{2} y^{3}-\frac{1}{1+9 x^{2}}\right) \frac{d x}{d y}+x^{3} y^{2}=0 $$

In Problems 9-16, solve the given differential equation. $$ \left(x^{2}+4\right) d y=(2 x-8 x y) d x $$

Each \(D E\) in Problems \(15-22\) is a Bernoulli equation. In Problems 15-20, solve the given differential equation by using an appropriate substitution. $$ x \frac{d y}{d x}+y=\frac{1}{y^{2}} $$

In Problems 1-22, solve the given differential equation by separation of variables. \(\frac{d S}{d r}=k S\) 16\. \(\frac{d Q}{d t}=k(Q-70)\)

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution. $$ y d x=\left(y e^{y}-2 x\right) d y $$

Solve the given differential equation by using an appropriate substitution. $$ \frac{d y}{d x}-y=e^{x} y^{2} $$

(a) Consider the direction field of the differential equation \(d y / d x=x(y-4)^{2}-2\), but do not use technology toobtain it. Describe the slopes of the lineal elements on the lines \(x=0, y=3, y=4\), and \(y=5\) (b) Consider the IVP \(d y / d x=x(y-4)^{2}-2, y(0)=y_{0}\), where \(y_{0}<4\). Can a solution \(y(x) \rightarrow \infty\) as \(x \rightarrow \infty\) ? Based on the information in part (a), discuss.

In Problems 9-16, solve the given differential equation. $$ \left(2 r^{2} \cos \theta \sin \theta+r \cos \theta\right) d \theta+\left(4 r+\sin \theta-2 r \cos ^{2} \theta\right) d r=0 $$

Each \(D E\) in Problems \(15-22\) is a Bernoulli equation. In Problems 15-20, solve the given differential equation by using an appropriate substitution. $$ \frac{d y}{d x}-y=e^{x} y^{2} $$

In Problems 1-22, solve the given differential equation by separation of variables. $$ \frac{d Q}{d t}=k(Q-70) $$

A thermometer ceading \(70^{\circ} \mathrm{F}\) is placed in an oven preheated to a constant t-mperahire. Turough a glass window in the oven door, an observer records that the thermometer read \(110^{\circ} \mathrm{F}\) after \(\frac{1}{2}\) mimute and \(145^{\circ} \mathrm{F}\) after 1 minute. How hot is the oven?

SIR Model A communicable disease is spread throughout a small community, with a fixed population of \(n\) people, by contact between infected individuals and people who are susceptible to the disease. Suppose initially that everyone is susceptible to the disease and that no one leaves the community while the epidemic is spreading. At time \(t\), let \(s(t), i(t)\), and \(r(t)\) denote, in tum, the number of people in the community (measured in hundreds) who are susceptible to the disease but not yet infected with it, the number of people who are infected with the disease, and the number of people who have recovered from the disease. Explain why the system of differential equations $$ \begin{aligned} &\frac{d s}{d t}=-k_{1} s i \\ &\frac{d i}{d t}=-k_{2} i+k_{1} s i \\ &\frac{d r}{d t}=k_{2} i \end{aligned} $$ where \(k_{1}\) (called the infection rate) and \(k_{2}\) (called the removal rate) are positive constants, is a reasonable mathematical model, commonly called a SIR model, for the spread of the epidemic throughout the community. Give plausible initial conditions associated with this system of equations. Show that the system implies that $$ \frac{d}{d t}(s+i+r)=0 $$

A differential equation governing the velocity \(v\) of a falling mass \(m\) subjected to air resistance proportional to the square of the instantaneous velocity is $$ m \frac{d v}{d t}=m g-k v^{2}, $$ where \(k>0\) is the drag coefficient. The positive direction is downward. (a) Solve this equation subject to the initial condition \(v(0)=v_{0}\) (b) Use the solution in part (a) to determine the limiting, or terminal, velocity of the mass. We saw how to determine the terminal velocity without solving the \(\mathrm{DE}\) in Problem 39 in Exercises \(2.1\). (c) If distance \(s\), measured from the point where the mass was released above ground, is related to velocity \(v\) by \(d s / d t=v(t)\), find an explicit expression for \(s(t)\) if \(s(0)=0\).

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution. $$ \cos x \frac{d y}{d x}+(\sin x) y=1 $$

Solve the given differential equation by using an appropriate substitution. $$ \frac{d y}{d x}=y\left(x y^{3}-1\right) $$

Determine whether the given differential equation is exact. If it is exact, solve it. $$ (\tan x-\sin x \sin y) d x+\cos x \cos y d y=0 $$

For a first-order DE \(d y / d x=f(x, y)\), a curve in the plane defined by \(f(x, y)=0\) is called a nullcline of the equation, since a lineal element at a point on the curve has zero slope. Use computer software to obtain a direction field over a rectangular grid of points for \(d y / d x=x^{2}-2 y\), and then superimpose the graph of the nullcline \(y=\frac{1}{2} x^{2}\) over the direction field. Discuss the behavior of solution curves in regions of the plane defined by \(y<\frac{1}{2} x^{2}\) and by \(y>\frac{1}{2} x^{2}\). Sketch some approximate solution curves. Try to generalize your observations.

In Problems 17 and 18, solve the given initial-value problem and give the largest interval \(I\) on which the solution is defined. $$ \sin x \frac{d y}{d x}+(\cos x) y=0, \quad y(7 \pi / 6)=-2 $$

Air Resistance A differential equation governing the velocity \(v\) of a falling mass \(m\) subjected to air resistance proportional to the square of the instantaneous velocity is $$ m \frac{d v}{d t}=m g-k v^{2} $$ where \(k>0\) is the drag coefficient. The positive direction is downward. (a) Solve this equation subject to the initial condition \(v(0)=v_{0}\). (b) Use the solution in part (a) to determine the limiting, or terminal, velocity of the mass. We saw how to determine the terminal velocity without solving the DE in Problem 39 in Exercises 2.1. (c) If distance \(s\), measured from the point where the mass was released above ground, is related to velocity \(v\) by \(d s / d t=v(t)\), find an explicit expression for \(s(t)\) if \(s(0)=0\).

In Problems 1-22, solve the given differential equation by separation of variables. $$ \frac{d P}{d t}=P-P^{2} $$

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution. $$ \cos ^{2} x \sin x \frac{d y}{d x}+\left(\cos ^{3} x\right) y=1 $$

Solve the given differential equation by using an appropriate substitution. $$ x \frac{d y}{d x}-(1+x) y=x y^{2} $$