Word Problems Examples

Translate the following English statement into a differential equation:

Jenna's savings account grows at a rate of $300 per year.

The two quantities described are savings (in $) and time (in years). Let S stand for savings and t for time in years. Then the "rate" at which Jenna's savings account grows is the same thing as the "derivative" of savings with respect to time, or

The statement says the rate of change of savings is $300 dollars per year. As a differential equation,

Translate the following English statement into a differential equation:

"The population is increasing at a rate proportional to the size of the population."

The statement is talking about the size of a population. Let's use the letter P to refer to the population size. Then the original statement

The population is increasing at a rate proportional to the size of the population.

becomes the more math-like statement

P is increasing at a rate proportional to P.

Since population size is a function of time, and this statement doesn't describe any variables besides population size and time, we'll assume that the rate described in the statement is

where t stands for time. The statement says this rate is proportional to P, which means equal to a constant multiplied by P. So we get this equation:

where k is the constant of proportionality.

Translate the following English statement into a differential equation:

The population is increasing at a rate of 2 percent per year.

This is similar to the previous problem. Let's use P for population size and t for time in years. It might be tempting to write

but this is wrong! This would mean the population grows by .02 people per year, which doesn't make any sense. Saying "the population is increasing at a rate of 2 percent" really means "the population is increasing at a rate of 2 percent of the current population," or a rate of 0.02P. The correct differential equation is

A population P has a birth rate of 1 percent and 200 people die every year. Find the overall rate of change of the population.

If the population has a birth rate of 1 percent, its rate of increase is

0.01P.

If 200 people die per year, the rate of decrease is 200. To get the overall rate of change , we take the birth rate and subtract off the death rate:

Julie makes $200 dollars per week and puts it in her bank account. After making the deposit, she spends 20 percent of the money in her bank account. Model this situation with a differential equation.

Julie makes $200 dollars per week, so her bank account S increases at a rate of 200. She spends 20 percent of the money, so S decreases at a rate of 0.2S. Her bank account changes at a rate of