Solve the following differential equations:

$$y^{\prime}=(y-3)^2 \ln t$$

Solve the given differential equations.

$$d y+y d x=e^{-x} d x$$

Solve the given differential equations.

$$\sin x \sec y d x=d y$$

Solve the given differential equations.

$$y^{\prime}+y \tan x=-\sin x$$

Solve the given differential equations.

$$y^{\prime}+2 y=\sin x$$

Evaluate the differential equations.

$$\sec x y^{\prime}-x y /(y+4)=0$$

In an experiment, a certain type of bacteria was being added to a culture at the rate of $e^{.03 t}+2$ thousand bacteria per hour. Suppose that the bacteria grow at a rate proportional to the size of the culture at time t, with constant of proportionality k=.45. Let P(t) denote the number of bacteria in the culture at time t. Find a differential equation satisfied by P(t).

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

$$6 y^{\prime}+t y=t$$

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.

$y^{\prime}=g(y), y(0)=-.5, y(0)=.5$, where g(y) is the function.

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

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

A certain piece of news is being broadcast to a potential audience of 200,000 people. Let f(t) be the number of people who have heard the news after t hours. Suppose that y=f(t) satisfies

$$y^{\prime}=.07(200,000-y), \quad y(0)=10 .$$

Describe this initial-value problem in words.

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

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

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

$$y^{\prime}=5+4 y-y^2, y(0)=1$$

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.