Modeling with First Order Equations

Main idea

Most first-order models come from one piece of bookkeeping: how fast an amount changes is how fast it comes in minus how fast it goes out. Mixing tanks, drug levels, populations, cooling and falling objects all start there.

{\htmlData{sym=0}{\frac{dQ}{dt}}} = (\text{rate in}) - (\text{rate out})

What the symbols mean

$\frac{dQ}{dt}$
Net rate
How fast the amount changes: inflow minus outflow.

Explanation

Modeling is simply turning a real situation, like salt flowing through a tank or a drug in the bloodstream, into a differential equation you can solve. It's how you go from a word problem to numbers you can actually predict with. All we're really saying is what you'd say about a bank account: the balance changes by what goes in minus what goes out. The formula $\frac{dQ}{dt} = (\text{rate in}) - (\text{rate out})$ is just three pieces: $\frac{dQ}{dt}$ (how fast the amount is changing), the rate in (how much arrives per unit of time) and the rate out (how much leaves per unit of time). Every model in this lesson is that sentence with specific numbers plugged in. It can look complex once the units pile up, but it's just a few simple parts, so bear with it.

Mixing tanks. If brine with concentration $c_{\text{in}}$ flows in at rate $r_{\text{in}}$ and the well-mixed solution leaves at $r_{\text{out}}$ , then

Q' = c_{\text{in}}r_{\text{in}} - \frac{Q}{V(t)}r_{\text{out}},

where $V(t)$ is the volume in the tank. Think about where each piece comes from. Salt arriving per minute is (grams per liter) times (liters per minute). Salt leaving works the same way, except the concentration going out is whatever is in the tank right now, $\frac{Q}{V}$ . That's why $Q$ shows up on the right side, and why these models usually come out linear, so the integrating factor from earlier handles them.

A common mistake. It's tempting to divide by the starting volume, because that's the number the problem gives you. That's only right when the flows match. If liquid comes in faster than it leaves, the tank fills, and the volume is $V(t)$ , not a constant. The challenge problem below is exactly that case.

Falling with drag. Newton's law with air resistance proportional to speed: $mv' = mg - kv$ . Gravity is the rate in, drag is the rate out. The velocity approaches the terminal velocity $\frac{mg}{k}$ , where the two cancel.

Growth and cooling. $P' = kP$ and $T' = -k(T - T_a)$ , which you've already solved.

Checking a model: units must match on both sides, and the long-term behavior should make physical sense.

See it

Brine at 2 g/L flows into a tank holding 50 L of fresh water at 3 L/min while 2 L/min drains out, so the volume grows along with the salt. Rate in minus rate out gives Q′ = 6 − 2Q/(50 + t). The concentration climbs toward the incoming 2 g/L but hasn't caught up when the tank fills at t = 50: it's only at 1.75 g/L.

Worked examples

Example 1

A 1000 L tank holds water with 50 g of salt. Brine with 2 g/L enters at 5 L/min, and the mixture leaves at 5 L/min. Find the salt over time.

1
Salt comes in at 2 g/L times 5 L/min, and leaves at the tank's concentration $\frac{Q}{1000}$ times 5 L/min. The flows match, so the volume stays 1000 L: $Q' = (2)(5) - \frac{Q}{1000}(5) = 10 - \frac{Q}{200}$ , $Q(0) = 50$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

In the tank example, how much salt is there after a very long time?

Problem 2+5 XP

Write $Q(t)$ for the tank example.

Problem 3+5 XP

A population grows by $P' = 0.03P$ . How long does it take to double?

Problem 4+5 XP

Find the terminal velocity for $m = 2$ kg, $k = 0.5$ kg/s, $g = 9.8$ m/s².

Problem 5+5 XP

Coffee cools by $T' = -0.1(T - 20)$ with $T(0) = 90$ . Find $T(10)$ .

Problem 6+5 XP

A vat holds $800$ L of water with $30$ kg of sugar dissolved in it. Starting at $t = 0$ , a sugar solution with $0.15$ kg/L flows in at $8$ L/min, and the well-stirred mixture drains at $8$ L/min. Set up the IVP for the sugar $x(t)$ in kg: enter the right side of $x' = \ldots$ as an expression in $x$ .

Problem 7+5 XP

A patient has about $6$ L of blood with $15$ mg of a drug in it. An IV adds the drug at $3$ mg/min, and the liver and kidneys clear $0.12$ L of blood per minute. With the drug evenly mixed, set up the IVP for the amount $A(t)$ in mg: enter the right side of $A' = \ldots$ as an expression in $A$ .

Problem 8+5 XP

A frozen turkey at $-18°$ C goes into a $180°$ C oven. Its temperature changes at a rate proportional to the difference between it and the oven, with constant $0.012$ per minute. Enter the right side of $T' = \ldots$ as an expression in $T$ .

Problem 9+5 XP

Which equation says: the rate of growth of a population is proportional to the average of the population and its cube root?

Problem 10+5 XP

Which equation says: the rate of change of an object's position is proportional to the square of the product of its position and its acceleration?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

A $100$ L tank initially holds $50$ L of pure water. Brine with $2$ g of salt per liter flows in at $3$ L/min, and the well-mixed solution drains at $2$ L/min. Find the amount of salt $Q(t)$ , in grams, while the tank is filling.

Where it's used

Pharmacokinetics (how drugs build up and clear), pollution in lakes, the charge on a capacitor and the spread of a rumor are all "rate in minus rate out" models.

IV Drip ConcentrationDiff Eq · Medicine Newton's Law of CoolingDiff Eq · Physics Paying Off a LoanDiff Eq · Economics & Finance

Intervals of Validity Equilibrium Solutions