Chapter 2

Find the critical points and phase portrait of the given autonomous first- order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the \(x y\) -plane determined by the graphs of the equilibrium solutions. $$ \frac{d y}{d x}=10+3 y-y^{2} $$

Each \(D E\) in Problems 23-30 is of the form given in (5). In Problems 23-28, solve the given differential equation by using an appropriate substitution. $$ \frac{d y}{d x}=\frac{1-x-y}{x+y} $$

In Problems 23-28, find an implicit and an explicit solution of the given initial-value problem. $$ \frac{d y}{d x}=\frac{y^{2}-1}{x^{2}-1}, \quad y(2)=2 $$

Solve the given initial-value problem. Give the largest interval \(I\) over which the solution is defined. $$ x y^{\prime}+y=e^{x}, \quad y(1)=2 $$

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

Solve the given initial-value problem. $$ \begin{aligned} &\left(y^{2} \cos x-3 x^{2} y-2 x\right) d x+\left(2 y \sin x-x^{3}+\ln y\right) d y=0 \\ &y(0)=e \end{aligned} $$

In Problems, find an implicit and an explicit solution of the given initial- value problem. \(x^{2} \frac{d y}{d x}=y-x y, \quad y(-1)=-1\)

Find the critical points and phase portrait of the given autonomous first- order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the \(x y\) -plane determined by the graphs of the equilibrium solutions. $$ \frac{d y}{d x}=y^{2}\left(4-y^{2}\right) $$

Each \(D E\) in Problems 23-30 is of the form given in (5). In Problems 23-28, solve the given differential equation by using an appropriate substitution. $$ \frac{d y}{d x}=\tan ^{2}(x+y) $$

In Problems 23-28, find an implicit and an explicit solution of the given initial-value problem. $$ x^{2} \frac{d y}{d x}=y-x y, \quad y(-1)=-1 $$

A skydiver is equipped with a stopwatch and an altimeter. She opens her parachute 25 seconds after exiting a plane flying at an altitude of \(20,000 \mathrm{ft}\) and observes that her altitude is \(14,800 \mathrm{ft}\). Assume that air resistance is proportional to the square of the instantaneous velocity, her initial velocity upon leaving the plane is zero, and \(g=32 \mathrm{ft} / \mathrm{s}^{2}\) (a) Find the distance \(s(t)\), measured from the plane, that the skydiver has traveled during free fall in time \(t\). (b) How far does the skydiver fall and what is her velocity at \(t=15 \mathrm{~s} ?\)

Solve the given initial-value problem. Give the largest interval \(I\) over which the solution is defined. $$ y \frac{d x}{d y}-x=2 y^{2}, \quad y(1)=5 $$

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

Solve the given initial-value problem. $$ \left(\frac{1}{1+y^{2}}+\cos x-2 x y\right) \frac{d y}{d x}=y(y+\sin x), \quad y(0)=1 $$

In Problems, find an implicit and an explicit solution of the given initial- value problem. \(\frac{d y}{d t}+2 y=1, \quad y(0)=\frac{5}{2}\)

Find the critical points and phase portrait of the given autonomous first- order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the \(x y\) -plane determined by the graphs of the equilibrium solutions. $$ \frac{d y}{d x}=y(2-y)(4-y) $$

Air containing \(0.06 \%\) carbon dioxide is pumped into a room whose volume is \(8000 \mathrm{ft}^{3}\). The air is pumped in at a rate of \(2000 \mathrm{ft}^{3} / \mathrm{min}\), and the circulated air is then pumped out at the same rate. If there is an initial concentration of \(0.2 \%\) carbon dioxide, determine the subsequent amount in the room at any time. What is the concentration at 10 minutes? What is the steady-state, or equilibrium, concentration of carbon dioxide?

Each \(D E\) in Problems 23-30 is of the form given in (5). In Problems 23-28, solve the given differential equation by using an appropriate substitution. $$ \frac{d y}{d x}=\sin (x+y) $$

In Problems 23-28, find an implicit and an explicit solution of the given initial-value problem. $$ \frac{d y}{d t}+2 y=1, \quad y(0)=\frac{5}{2} $$

A large tank is partially filled with 100 gallons of fluid in which 10 pounds of salt is dissolved. Brine containing \(\frac{1}{2}\) pound of salt per gallon is pumped into the tank at a rate of \(6 \mathrm{gal} / \mathrm{min}\). The well-mixed solution is then pumped out at a slower rate of \(4 \mathrm{gal} / \mathrm{min}\). Find the number of pounds of salt in the tank after 30 minutes.

Solve the given initial-value problem. Give the largest interval \(I\) over which the solution is defined. $$ L \frac{d i}{d t}+R i=E ; \quad i(0)=i_{0}, L, R, E, \text { and } i_{0} \text { constants } $$

Solve the given differential equation by using an appropriate substitution. $$ \frac{d y}{d x}=2+\sqrt{y-2 x+3} $$

Find the value of \(k\) so that the given differential equation is exact. $$ \left(y^{3}+k x y^{4}-2 x\right) d x+\left(3 x y^{2}+20 x^{2} y^{3}\right) d y=0 $$

Find the critical points and phase portrait of the given autonomous first- order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the \(x y\) -plane determined by the graphs of the equilibrium solutions. $$ \frac{d y}{d x}=y \ln (y+2) $$

Solve the differential equation $$ \frac{d y}{d x}=-\frac{y}{\sqrt{s^{2}-y^{2}}} $$ of the tractrix. See Problem 28 in Exercises 1.3. Assume that the initial point on the \(y\)-axis is \((0,10)\) and that the length of the rope is \(x=10 \mathrm{ft}\).

Each \(D E\) in Problems 23-30 is of the form given in (5). In Problems 23-28, solve the given differential equation by using an appropriate substitution. $$ \frac{d y}{d x}=2+\sqrt{y-2 x+3} $$

Solve the given initial-value problem. Give the largest interval \(I\) over which the solution is defined. $$ L \frac{d i}{d t}+R i=E ; \quad i(0)=i_{0}, L, R, E, \text { and } i_{0} \text { constants } $$

Solve the given differential equation by using an appropriate substitution. $$ \frac{d y}{d x}=1+e^{y-x+5} $$

Find the value of \(k\) so that the given differential equation is exact. $$ \left(6 x y^{3}+\cos y\right) d x+\left(2 k x^{2} y^{2}-x \sin y\right) d y=0 $$

In Problems, find an implicit and an explicit solution of the given initial- value problem. \(\left(1+x^{4}\right) d y+x\left(1+4 y^{2}\right) d x=0, \quad y(1)=0\)

Find the critical points and phase portrait of the given autonomous first- order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the \(x y\) -plane determined by the graphs of the equilibrium solutions. $$ \frac{d y}{d x}=\frac{y e^{y}-9 y}{e^{y}} $$

In Example 5 the size of the tank containing the salt mixture was not given. Suppose, as in the discussion following Example 5, that the rate at which brine is pumped into the tank is \(3 \mathrm{gal} / \mathrm{min}\) but that the well-stirred solution is pumped out at a rate of \(2 \mathrm{gal} / \mathrm{min}\). It stands to reason that since brine is accumulating in the tank at the rate of \(1 \mathrm{gal} / \mathrm{min}\), any finite tank must eventually overflow. Now suppose that the tank has an open top and has a total capacity of 400 gallons. (a) When will the tank overflow? (b) What will be the number of pounds of salt in the tank at the instant it overflows? (c) Assume that although the tank is overflowing, the brine solution continues to be pumped in at a rate of \(3 \mathrm{gal} / \mathrm{min}\) and the well- stirred solution continues to be pumped out at a rate of \(2 \mathrm{gal} / \mathrm{min}\). Devise a method for determining the number of pounds of salt in the tank at \(t=150 \mathrm{~min}\). (d) Determine the number of pounds of salt in the tank as \(t \rightarrow \infty\). Does your answer agree with your intuition? (e) Use a graphing utility to plot the graph \(A(t)\) on the interval \([0,500)\).

Each \(D E\) in Problems 23-30 is of the form given in (5). In Problems 23-28, solve the given differential equation by using an appropriate substitution. $$ \frac{d y}{d x}=1+e^{y-x+5} $$

A 30-volt electromotive force is applied to an \(L R\) -series circuit in which the inductance is \(0.1\) henry and the resistance is 50 ohms. Find the current \(i(t)\) if \(i(0)=0\). Determine the current as \(t \rightarrow \infty\).

Solve the given initial-value problem. Give the largest interval \(I\) over which the solution is defined. $$ (x+1) \frac{d y}{d x}+y=\ln x, \quad y(1)=10 $$

Solve the given initial-value problem. $$ \frac{d y}{d x}=\cos (x+y), \quad y(0)=\pi / 4 $$

Verify that the given differential equation is not exact. Multiply the given differential equation by the indicated integrating factor \(\mu(x, y)\) and verify that the new equation is exact. Solve. $$ (-x y \sin x+2 y \cos x) d x+2 x \cos x d y=0 ; \quad \mu(x, y)=x y $$

$$ \text { In Problems } 29 \text { and } 30 \text {, solve the given initial- value problem. } $$ $$ \frac{d y}{d x}=\cos (x+y), \quad y(0)=\pi / 4 $$

In Problems 29 and 30, proceed as in Example 5 and find an explicit solution of the given initial-value problem. $$ \frac{d y}{d x}=y e^{-x^{2}}, y(4)=1 $$

Solve the given initial-value problem. Give the largest interval \(I\) over which the solution is defined. $$ y^{\prime}+(\tan x) y=\cos ^{2} x, \quad y(0)=-1 $$

Solve the given initial-value problem. $$ \frac{d y}{d x}=\frac{3 x+2 y}{3 x+2 y+2}, \quad y(-1)=-1 $$

Verify that the given differential equation is not exact. Multiply the given differential equation by the indicated integrating factor \(\mu(x, y)\) and verify that the new equation is exact. Solve. $$ \left(x^{2}+2 x y-y^{2}\right) d x+\left(y^{2}+2 x y-x^{2}\right) d y=0 ; \mu(x, y)=(x+y)^{-2} $$

According to Stefan's law of radiation, the absolute temperature \(T\) of a body cooling in a medium at constant temperature \(T_{m}\) is given by $$ \frac{d T}{d t}=k\left(T^{4}-T_{m}^{4}\right) $$ where \(k\) is a constant. Stefan's law can be used over a greater temperature range than Newton's law of cooling. (a) Solve the differential equation. (b) Show that when \(T-T_{m}\) is small compared to \(T_{m}\) then Newton's law of cooling approximates Stefan's law. [Hint: Think binomial series of the right- hand side of the DE.]

$$ \text { In Problems } 29 \text { and } 30 \text {, solve the given initial- value problem. } $$ $$ \frac{d y}{d x}=\frac{3 x+2 y}{3 x+2 y+2}, \quad y(-1)=-1 $$

Solve the given initial-value problem. Give the largest interval \(I\) over which the solution is defined. $$ \left(\frac{e^{-2 \sqrt{x}}-y}{\sqrt{x}}\right) \frac{d x}{d y}=1, \quad y(1)=1 $$

Explain why it is always possible to express any homogeneous differential equation \(M(x, y) d x+N(x, y) d y=0\) in the form $$ \frac{d y}{d x}=F\left(\frac{y}{x}\right) $$ You might start by proving that $$ M(x, y)=x^{a} M(1, y / x) \text { and } N(x, y)=x^{\alpha} N(1, y / x) $$

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

Consider the autonomous \(\mathrm{DE} d y / d x=(2 / \pi) y-\sin y\) 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.

An \(L R\)-series circuithas a variable inductor with the inductance defined by $$ L(t)= \begin{cases}1-\frac{1}{10} t, & 0 \leq t<10 \\ 0, & t \geq 10 .\end{cases} $$ Find the current \(i(t)\) if the resistance is \(0.2 \mathrm{ohm}\), the impressed voltage is \(E(t)=4\), and \(i(0)=0\). Graph \(i(t)\).

A 100-volt electromotive force is applied to an \(R C\)-series circuit in which the resistance is 200 ohms and the capacitance is \(10^{-4}\) farad. Find the charge \(q(t)\) on the capacitor if \(q(0)=0\). Find the current \(i(t)\).