Differential Equations Exercises

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We have a population P and time t measured in years. The population increases at a rate of 1,000 people per year, which means

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The two variables are the number of bunnies B and the time t. The statement

The number of bunnies is increasing at a rate proportional to the number of bunnies can be condensed into the statement B is increasing at a rate proportional to B. As a differential equation,

where k is the constant of proportionality.

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There are a couple of ways to do this problem. The two variables involved are money M and time t. We know that Tamara's money is decreasing at a rate of $40 dollars per week. If we let t be measured in weeks we get the differential equation

Alternatively, we could calculate that Tamara spends 40 × 52 = 2080 dollars per year, let t be measured in years, and write

In either case, we need the negative sign because Tamara's money is decreasing, meaning the rate of change of her money with respect to time is negative.

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We're looking at the rate of change of J (the number of pieces of junk mail Ben receives) with respect to t (time). It probably makes the most sense to measure time in days, in which case we get the differential equation

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The variables in question are the temperature of the cookies T (measured in °F) and time t. The original statement

The temperature of the cookies increases at a rate proportional to the difference between the temperature of the cookies and the temperature of the oven

can be simplified:

T increases at a rate proportional to the difference between T and the temperature of the oven.Since we know the temperature of the oven is 375°F, we know that is proportional to ( T – 375) or ( 365 – T ) (either of these is ok). So the final differential equation can be either

or

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Since the population of bunnies B is increasing, the rate of change must be positive. Since B is positive, k must also be positive so that the product kB is positive.

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Since the population of geese G is decreasing, the rate of change must be negative. Since G is positive, in order for the product kG to be negative, k must be negative.

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If Q is positive and k is negative, then

-kQ = -(negative)(positive)

is positive. This means

is positive, so Q is increasing.

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If cookies are baking, their temperature is increasing, so

must be positive. Since the cookies start out cooler than 375, the quantity (375 – T) is positive. In order for the product k(375 – T) to be positive, k must also be positive.

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If coffee is cooling, its temperature is decreasing, so

must be negative. Since the coffee is warmer than 68° the quantity (68 – T) is negative. In order for the product

k(68 – T)

to be negative, the constant k must be positive.

Some of the problems we've had so far have been way too simplified to be real. People don't just spend $40 per week. In real life, a bank account has some money going in and some money going out.

A population doesn't just increase. In real life, some people are born and some people die.

For most quantities we model with differential equations, we can think of the rate of change in two separate pieces:

• the rate of increase (birth rate, rate of saving)
• the rate of decrease (death rate, rate of spending)

To get the overall rate of change, we combine the rates of increase and decrease:

(overall rate of change) = (rate of increase) – (rate of decrease).

Remember that the units of the derivative are the units of the dependent variable over the units of the independent variable.

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Let W stand for the amount of water in gallons and t for the time in minutes. Then

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Let W stand for the number of wolves and t stand for time in years. The birth rate is 5 percent of the current population, or .05W. The death rate is 200. Putting it together,

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Let S be the amount of money in Donna's account and t be time in months. Every month Donna's account gains 2 percent or .02S in interest, and also gains $100. therefore the rate at which her savings account grows is

Since no money is taken out in this problem, we don't need to worry about the rate at which money leaves.

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Let S be the amount of sand on the beach, measured in cubic feet. Let t be time in years. The rate at which sand accumulates is 150 cubic feet per year. The rate at which sand is carried away is 10 percent of the current amount of sand per year, or .1S. Putting these together,

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Let W be number of wasps and t be time in years. Then

If we look at the 10 percent birth rate and 9 percent death rate, we see that the wasp population should be increasing by 1 percent per year. Then people swat another 300 wasps per year, so we need to subtract another 300 from the rate.