7.10 Constructing differential equations — Learn

Prerequisites: the derivative as a rate of change from Section 7.1; differentiating standard functions from Section 7.3; reading a word problem and choosing a letter for each quantity.

By the end of this section you can: turn a description of a rate into a differential equation; use proportionality and a sign to model growth, decay, cooling, motion, price and demand; and eliminate a constant from a family of curves to form a differential equation.

1. From words to an equation

A differential equation is an equation that links a derivative to the quantities it describes. To build one, follow three steps.

1. Name the quantity and its rate. For a quantity $P$ that changes with time $t$, the rate is $\dfrac{dP}{dt}$.
2. Read the relationship. "Proportional to" introduces a positive constant $k$.
3. Get the sign right. "Grows" and "increases" need a positive term; "falls", "cools" and "decays" need a minus sign.

Example 1. A population of bacteria grows at a rate proportional to the population. Write a differential equation for the population $P$ after $t$ hours.

The rate is $\dfrac{dP}{dt}$. It is proportional to $P$, with a positive constant:

Example 2. A cup of tea cools. The rate at which its temperature falls is proportional to the difference between the temperature and the room temperature, $20^\circ$C. Write a differential equation for the temperature $\theta$ after $t$ minutes.

The rate is $\dfrac{d\theta}{dt}$. The temperature difference is $\theta - 20$. The temperature falls, so the rate is negative:

The minus sign is the important part: the tea cools only while it is warmer than the room.

Example 3. A stone falls under gravity. Its velocity $v$ increases at a rate proportional to the constant $g$, directed downwards. Write a differential equation for $v$ after $t$ seconds.

The rate of change of velocity is acceleration. The velocity increases downwards, so with downwards taken as positive,

If the stone also meets air resistance proportional to its speed, the equation becomes

Example 4. The price $P$ of a product falls at a rate proportional to the amount by which the price exceeds $10$. Write a differential equation for $P$.

The price falls, so the rate is negative, and it is proportional to $P - 10$:

2. Forming a differential equation from a family of curves

A differential equation also describes a whole family of curves. Differentiate the family and eliminate any constant.

Example 5. The curve $y = Ae^{2x}$ has a constant $A$. Form a differential equation that does not contain $A$.

Differentiate:

But $Ae^{2x}$ is $y$, so

The differential equation describes every curve in the family, whatever the value of $A$.

Example 6. The curve $y = x^3 + C$ has a constant $C$. Form a differential equation that does not contain $C$.

Differentiate once:

The constant $C$ has vanished, so $\dfrac{dy}{dx} = 3x^2$ is the differential equation. Every curve in the family has the same gradient at a given $x$.

Watch out. Three mistakes are common when forming differential equations.

Forgetting the minus sign for a decrease. "Cools" and "decays" mean the rate is negative. In Example 2 the equation $\dfrac{d\theta}{dt} = k(\theta - 20)$ would make the tea hotter, which is the wrong model.

Writing "proportional to" as an equals sign without the constant. The phrase introduces an unknown constant $k$. Write $kP$, not $P$.

Keeping the wrong variable in the rate. The rate of change of $P$ with respect to $t$ is $\dfrac{dP}{dt}$. Do not write $\dfrac{dt}{dP}$, which is the reciprocal. Re-check: after you write the equation, read it back in words and confirm that the sign and the rate match the sentence.

You should now be able to

• name the rate in a description and write it as a derivative;
• use a constant $k$ for "proportional to";
• use a minus sign for cooling, decay and decrease;
• model population growth, cooling, motion, and price and demand;
• form a differential equation from a family of curves by eliminating a constant.