Modeling with differential equations

Suppose you have a bank account that grows by 5% every year. Let $A(t)$ be the amount of money in the account in year $t$.

1. What is the rate of change of $A$ with respect to $t$?

2. Suppose that you are also withdrawing $10,000 per year. Write a differential equation that expresses the total rate of change of $A$.

3. Sketch a slope field for this differential equation, find any equilibrium solutions, and identify them a…

A dose of morphine is absorbed from the bloodstream of a patient at a rate proportional to the amount in the bloodstream.

1. Write a differential equation for $M(t)$, the amount of morphine in the patient's bloodstream, using $k$ as the constant proportionality.

2. Assuming that the initial dose of morphine is $M_0$, solve the initial value problem to find $M(t)$. Use the fact that the half-life for the absorption of mor…

Congratulations, you just won the lottery! In one option presented to you, you will be paid one million dollars a year for the next 25 years. You can deposit this money in an account that will earn 5% each year.

1. Set up a differential equation that describes the rate of change in the amount of money in the account. Two factors cause the amount to grow—first, you are depositing one millon dollars per…

When a skydiver jumps from a plane, gravity causes her downward velocity to increase at the rate of $g\approx 9.8$ meters per second squared. At the same time, wind resistance causes her velocity to decrease at a rate proportional to the velocity.

1. Using $k$ to represent the constant of proportionality, write a differential equation that describes the rate of change of the skydiver's velocity.

2. Find any equilibrium…

During the first few years of life, the rate at which a baby gains weight is proportional to the reciprocal of its weight.

1. Express this fact as a differential equation.

2. Suppose that a baby weighs 8 pounds at birth and 9 pounds one month later. How much will he weigh at one year?

3. Do you think this is a realistic model for a long time?

Suppose that you have a water tank that holds 100 gallons of water. A briny solution, which contains 20 grams of salt per gallon, enters the tank at the rate of 3 gallons per minute.

At the same time, the solution is well mixed, and water is pumped out of the tank at the rate of 3 gallons per minute.

1. Since 3 gallons enter the tank every minute and 3 gallons leave every minute, what can you conclude…