Sketch the solutions of the differential equations in Exercise given below. In each case, also indicate the constant solutions.

$$y^{\prime}=y^2+y, y(0)=-\frac{1}{3}$$

Let f(t) be the balance in a savings account at the end of t years. Suppose that y=f(t) satisfies the differential equation $y^{\prime}=.04 y+2000$. (a) If after 1 year the balance is $10,000, is it increasing or decreasing at that time? At what rate is it increasing or decreasing at that time? (b) Write the differential equation in the form$y^${\prime}$=k(y+M)$. (c) Describe this differential equation in words.

One or more initial conditions are given for the differential equation in the this exercise. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solutions. Include a yz-graph if one is not already provided. Always indicate the constant solutions on the ty-graph whether they are mentioned or not.

The graph of $z=-\frac{1}{2} y \ln (y / 30)$ has relative maximum point at $y \approx 11.0364$ and y-intercept at y-30; y(0)-1, y(0)=20, and y(0)=40. Sketch the solutions to the Gompertz growth equation

$$\frac{d y}{d t}=-\frac{1}{2} y \ln \frac{y}{30} .$$

Consider a certain commodity that is produced by many companies and purchased by many other firms. Over a relatively short period there tends to be an equilibrium price $p_0$ per unit of the commodity that balances the supply and the demand. Suppose that, for some reason, the price is different from the equilibrium price. The Evans price adjustment model says that the rate of change of price with respect to time is proportional to the difference between the actual market price p and the equilibrium price. Write a differential equation that expresses this relation. Sketch two or more solution curves.

Solve the following differential equations:

$$y^2 y^{\prime}=\tan t$$

The differential equation $y^{\prime}=2 t y+e^{t^2}, y(0)=5$, has solution $y=(t+5) e^{t^2}$. In the following table, fill in the second row with the values obtained from the use of a numerical method and the third row with the actual values calculated from the solution. What is the greatest difference between corresponding values in the second and third rows?

$t_i$	0	.2	.4	.6	.8	1	1.2	1.4	1.6	1.8	2
$y_i$	5
y	5	5.412									382.2

Derive the formula for the population in Example mentioned, if the population in 1995 was 2 million. (The formula is given following the solution of the said Example.)

The differential equation $y^{\prime}=e^t-2 y, y(0)=1$, has solution $y=\frac{1}{3}\left(2 e^{-2 t}+e^t\right)$. In the following table, fill in the second row with the values obtained from the use of a numerical method and the third row with the actual values calculated from the solution. What is the greatest difference between corresponding values in the second and third rows?

$t_i$	0	.25	.50	.75	1	1.25	1.5	1.75	2
$y_i$	1
y	1	.8324							2.4752

A body was found in a room when the room's temperature was $70^{\circ} \mathrm{F}$. Let f(t) denote the temperature of the body t hours from the time of death. According to Newton's law of cooling, f satisfies a differential equation of the form

$$y^{\prime}=k(T-y) .$$

(a) Find T.

(b) After several measurements of the body's temperature, it was determined that when the temperature of the body was 80 degrees it was decreasing at the rate of 5 degrees per hour. Find k.

(c) Suppose that at the time of death the body's temperature was about normal, say $98^{\circ} \mathrm{F}$. Determine f(t).

(d) When the body was discovered, its temperature was $85^{\circ} \mathrm{F}$. Determine how long ago the person died.

In Exercise given below, solve the given equation using an integrating factor. Take t>0.

$$(1+t) y^{\prime}+y=-1$$

Let f(t) be the balance in a savings account at the end of t years, and suppose that y=f(t) satisfies the differential equation $y^{\prime}=.05 y-10,000$. (a) If after 1 year the balance is $150,000, is it increasing or decreasing at that time? At what rate is it increasing or decreasing at that time? (b) Write the differential equation in the form$y^${\prime}$=k(y-M)$ (c) Describe this differential equation in words.

Solve the following differential equations:

$$y^{\prime}=\frac{\ln t}{t y}$$

In economic theory, the following model is used to describe a possible capital investment policy. Let f(t) represent the total invested capital of a company at time t. Additional capital is invested whenever f(t) is below a certain equilibrium value E, and capital is withdrawn whenever f(t) exceeds E. The rate of investment is proportional to the difference between f(t) and E. Construct a differential equation whose solution is f(t) and sketch two or three typical solution curves.

Sketch the solutions of the differential equations in Exercise given below. In each case, also indicate the constant solutions.

$$y^{\prime}=2 \cos y, y(0)=0$$