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Change of variables in a Bernoulli equation The equation $y^{\prime}(t)+a y=b y^{p},$ where $a, b,$ and $p$ are real numbers, is called a Bernoulli equation. Unless $p=1$, the equation is nonlinear and would appear to be difficult to solve except for a small miracle. By making the change of variables $v(t)=(y(t))^{1-p},$ the equation can be made linear. Carry out the following steps.a. Letting $v=y^{1-p},$ show that $y^{\prime}(t)=\frac{y(t)^{p}}{1-p} v^{\prime}(t)$b. Substitute this expression for $y^{\prime}(t)$ into the differential equation and simplify to obtain the new (linear) equation $v^{\prime}(t)+a(1-p) v=b(1-p),$ which can be solved using the methods of this section. The solution $y$ of the original equation can then be found from $v$

Cooling time Suppose an object with an initial temperature of $T_{0} > 0$ is put in surroundings with an ambient temperature of $A$ where $A < \frac{T_{0}}{2} .$ Let $t_{1 / 2}$ be the time required for the object to cool to $\frac{T_{0}}{2}$a. Show that $t_{1 / 2}=-\frac{1}{k} \ln \left[\frac{T_{0}-2 A}{2\left(T_{0}-A\right)}\right]$b. Does $t_{1 / 2}$ increase or decrease as $k$ increases? Explain.c. Why is the condition $A < \frac{T_{0}}{2}$ needed?

A special class of first-order linear equations have the form $a(t) y^{\prime}(t)+a^{\prime}(t) y(t)=f(t),$ where a and fare given functions of t. Notice that the left side of this equation can bewritten as the derivative of a product, so the equation has the form$$a(t) y^{\prime}(t)+a^{\prime}(t) y(t)=\frac{d}{d t}(a(t) y(t))=f(t)$$Therefore, the equation can be solved by integrating both sides with respect to $t .$ Use this idea to solve the following initial value problems.$$t^{3} y^{\prime}(t)+3 t^{2} y=\frac{1+t}{t}, y(1)=6$$

Consider the following differential equations. A detailed direction field is not needed.a. Find the solutions that are constant, for all $t \geq 0$ (the equilibrium solutions).b. In what regions are solutions increasing? Decreasing?c. Which initial conditions $y(0)=A$ lead to solutions that are increasing in time? Decreasing?d. Sketch the direction field and verify that it is consistent with parts $(a)-(c)$$$y^{\prime}(t)=(y-1)(1+y)$$

Plot a direction field for the following differential equation with a graphing utility. Then find the solutions that are constant and determine which initial conditions $y(0)=A$ lead to solutions that are increasing in time.$$y^{\prime}(t)=(y-1) \sin \pi t, \quad 0 \leq t \leq \pi, 0 \leq y \leq 2$$

Match each differential equation to the correct direction field.CAN'T COPY THE GRAPH$$y^{\prime}=\sqrt{x^{2}+y^{2}}$$

Apply Euler's method with $h=0.1$ to the initial value problem $y^{\prime}=y^{2}-1, y(0)=3$ and estimate $y(0.5) .$ Repeat with $h=0.05$ and $h=0.01 .$ In general, Euler's method is more accurate with smaller $h$ -values. Conjecture how the exact solution behaves in this example. (This is explored further in exercises $34-36 .)$

Find the exact solutions in exercises 13 and $14,$ and compare $y(1)$ and $y(2)$ to the Euler's method approximations.

Two steps of Euler's method For the following initial value problems, compute the first two approximations $u_{1}$and$u_{2}$ given by Euler's method using the given time step.$$y^{\prime}(t)=t+y, y(0)=4 ; \Delta t=0.5$$

Solve the following initial walue problems. When possible, give the solution as an explicit $$y^{\prime}(t)=\frac{\cos ^{2} t}{2 y}, y(0)=-2$$

Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to indicate which curve corresponds to the solution of the initial value problem.$$y^{\prime}(x)=\frac{1+x}{2-y}, y(1)=1$$

Determine whether the following equations are separable. If so, solve the initial value problem$$\frac{d y}{d t}=t y+2, y(1)=$$

Place these levels of ecological study in order from the least to the most comprehensive: community ecology, ecosystem ecology, organismal ecology, population ecology.

The population $p(t)$ (in millions) of the United States $t$ years after the year 1900 is shown in the figure. Approximately when (in what year) was the U.S. population growing most slowly between 1925 and 20207 Estimate the growth rate in that year.(GRAPH CAN'T COPY)

The deer population in Maine from 1986 to 2006 is approximated in the graph below. a) In what years was the deer population in Maine at or above 250,000 ?b) In what years was the deer population at 200,000 ?c) In what year was the deer population highest?d) In what years was the deer population lowest?

For an equation in $x$ and $y,$ if substitution of $a$ for $x$ and $b$ for $y$ satisfics the equation, then the point $(a, b)$ is a _____.

Show that $c_{1} \cos k t+c_{2} \sin k t=A \sin (k t+\delta),$ where$A=\sqrt{c_{1}^{2}+c_{2}^{2}}$ and $\tan \delta=\frac{c_{1}}{c_{2}} .$ We call $A$ the amplitude and8 the phase shift. Use this identity to find the amplitude and phase shift of the solution of $y^{\prime \prime}+9 y=0, y(0)=3$ and $y^{\prime}(0)=-6$

Write the second-order equation as a system of first-order equations.$$x y^{\prime \prime}+3\left(y^{\prime}\right)^{2}=y+2 x$$

Find and interpret all equilibrium points for the predator-prey model.$$\left\{\begin{array}{l}x^{\prime}=0.2 x-0.2 x^{2}-0.4 x y \\ y^{\prime}=-0.1 y+0.2 x y\end{array}\right.$$

Let $w$ be the number of worms (in millions) and $r$ the number of robins (in thousands) living on an island. Suppose $w$ and $r$ satisfy the following differential equations, which correspond to the slope field in Figure 9.44.(FIGURE CANNOT COPY)On the same axes, graph $w$ and $r$ (the worm and the robin populations) against time. Use initial values of 1.5 for $w$ and 1 for $r .$ You may do this without units for $t$.

Give the rates of growth of two populations, $x$ and $y,$ measured in thousands.(a) Describe in words what happens to the population of each species in the absence of the other.(b) Describe in words how the species interact with one another. Give reasons why the populations might behave as described by the equations. Suggest species that might interact in that way.$$\begin{aligned}&\frac{d x}{d t}=0.2 x\\&\frac{d y}{d t}=0.4 x y-0.1 y\end{aligned}$$

Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave upward or concave downward, intercepts where possible, and asymptotes where applicable. (Hint: In Exercise $21,$ use the result of Exercise $1 .$ In Exercises $25-27,$ recall from Exercise 66 in the section on Limits that $$\lim _{x \rightarrow \infty} x^{n} e^{-x}=0 .$$ )$$f(x)=\frac{-8}{x^{2}-6 x-7}$$

Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave upward or concave downward, intercepts where possible, and asymptotes where applicable. (Hint: In Exercise $21,$ use the result of Exercise $1 .$ In Exercises $25-27,$ recall from Exercise 66 in the section on Limits that $$\lim _{x \rightarrow \infty} x^{n} e^{-x}=0 .$$ )$$f(x)=16 x+\frac{1}{x^{2}}$$

Graph each function, considering the domain, critical points, symmetry, regions where the function is increasing or decreasing, inflection points, regions where the function is concave upward or concave downward, intercepts where possible, and asymptotes where applicable. (Hint: In Exercise $21,$ use the result of Exercise $1 .$ In Exercises $25-27,$ recall from Exercise 66 in the section on Limits that $$\lim _{x \rightarrow \infty} x^{n} e^{-x}=0 .$$ )$$f(x)=x^{4}-24 x^{2}+80$$

Suppose that a business has an income stream of $$ P(t) .$ The present value at interest rate $r$ of this income for the next $T$ years is $\int_{0}^{t} P(t) e^{-r t} d t .$ Compare the present values at $5 \%$ for three people with the following salaries for 3 years: $\begin{aligned}&\text { A: } \quad P(t)=60,000 ; \quad \text { B: } \quad P(t)=60,000+3000 t ; \quad \text { and }\\&\mathbf{C}: P(t)=60,000 e^{0.05 t} \end{aligned}$

In example $2.4,$ find and graph the solution passing through (0,0)

Use equation ( 2.6 ) to help solve the IVP.$$y^{\prime}=3 y(2-y), y(0)=1$$

A revenue model The owner of a clothing store understands that the demand for shirts decreases with the price. In fact, she has developed a model that predicts that at a price of $\$ x$ per shirt, she can sell $D(x)=40 e^{-x / 50}$ shirts in a day. It follows that the revenue (total money taken in) in a day is $R(x)=x D(x)(\$ x / \text { shirt } \cdot D(x) \text { shirts }) .$ What price should the owner charge to maximize revenue?

Free fall (adapted from Putnam Exam, 1939 ) An object moves freely in a straight line except for air resistance, which is proportional to its speed; this means its acceleration is $a(t)=-k v(t)$ The speed of the object decreases from $1000 \mathrm{ft} / \mathrm{s}$ to $900 \mathrm{ft} / \mathrm{s}$ over a distance of $1200 \mathrm{ft}$. Approximate the time required for this deceleration to occur. (Exercise 38 may be useful.)

Constant doubling time Prove that the doubling time for an exponentially increasing quantity is constant for all time.

For each system of equations in Example $2,$ write a differential equation involving $d y / d x .$ Use a computer or calculator to draw the slope field for $x, y>0 .$ Then draw the trajectory through the point $x=3, y=1$.

(a) Consider the slope field for $d y / d x=x y .$ What is the slope of the line segment at the point (2,1)$?$ At (0,2)$?$ At (-1,1)$?$ At (2,-2)$?$(b) Sketch part of the slope field by drawing line segments with the slopes calculated in part (a).

Sketch three solution curves for each of the slope fields in Figure 9.17. (FIGURE CAN'T COPY)