Separable differential equations

Suppose that the population of a town is growing continuously at an annual rate of 3% per year.

1. Let $P(t)$ be the population of the town in year $t$. Write a differential equation that describes the annual growth rate.

2. Find the solutions of this differential equation.

3. If you know that the town's population in year 0 is 10,000, find the population $P(t)$.

4. How long does it take for the population to double? This ti…

Suppose that a cup of coffee is initially at a temperature of $105^\circ$ F and is placed in a $75^\circ$ F room. If $T$ is the temperature of the coffee in degrees Fahrenheit at time $t$ in minutes, Newton's law of cooling says that , where $k$ is a constant of proportionality.

1. Suppose you measure that the coffee is cooling at one degree per minute at the time the coffee is brought into the room. Use the differential equa…

Solve each of the following differential equations or initial value problems.

1. $\frac{dy}{dt} - (2-t) y = 2-t$

2. $\frac{1}{t}\frac{dy}{dt} = e^{t^2-2y}$

3. $y' = 2y+2$,  $y(0)=2$

4. $y' = 2y^2$,  $y(-1) = 2$

5. $\frac{dy}{dt} = \frac{-2ty}{t^2 + 1}$,  $y(0) = 4$

The mass of a radioactive sample decays at a rate that is proportional to its mass.

1. Express this fact as a differential equation for the mass $M(t)$ using $k$ for the constant of proportionality.

2. If the initial mass is $M_0$, find an expression for the mass $M(t)$.

3. The half-life of the sample is the amount of time required for half of the mass to decay. Knowing that the half-life of Carbon-14 is 5730 years, find the…

Consider the initial value problem

1. Find the solution of the initial value problem and sketch its graph.

2. For what values of $t$ is the solution defined?

3. What is the value of $y$ at the last time that the solution is defined?

4. By looking at the differential equation, explain why we should not expect to find solutions with the value of $y$ you noted in (c).

Suppose that a cylindrical water tank with a hole in the bottom is filled with water. The water, of course, will leak out and the height of the water will decrease. Let $h(t)$ denote the height of the water. A physical principle called Torricelli's Law implies that the height decreases at a rate proportional to the square root of the height.

1. Express this fact using $k$ as the constant of proportionality.

2. S…