What is a Differential Equation?

Differential Equation (DE): An equation that relates a function to one or more of its derivatives. A first-order DE involves only the first derivative $\frac{d y}{d x}$ .

In the real world, many natural phenomena , population growth, cooling of objects, radioactive decay , are described not by simple equations, but by relationships between a quantity and its rate of change. This is precisely what a differential equation captures.

A general solution to a DE contains an arbitrary constant $C$ , representing a family of curves. A particular solution is found when an initial condition (a known point on the solution curve) is used to determine $C$ .

Initial Condition: A known value $y (x_{0}) = y_{0}$ that allows us to determine the arbitrary constant $C$ in the general solution, yielding a unique particular solution.

Note

In AHL 5.14, the primary focus is on setting up a DE from a written description and solving it by separating variables. You need to be comfortable translating a real-world scenario into a mathematical DE before you can solve it.

Setting Up a Differential Equation

Before solving a DE, you must construct it from a problem description. This is often the hardest step for students. The key is to identify what quantity is changing, what it is changing with respect to, and what governs the rate of change.

General Strategy:

Define your variables clearly (e.g., let $y$ = population at time $t$ ).
Express the rate of change in words, then translate to $\frac{d y}{d t}$ or $\frac{d y}{d x}$ .
Write the right-hand side as a function of the current variables.
Note any initial condition given.

Common phrasing and their mathematical translations:

Words	DE Form
"rate of change is proportional to $y$ "	$\frac{d y}{d t} = k y$
"rate of change is proportional to $(y - A)$ "	$\frac{d y}{d t} = k (y - A)$
"rate of change is proportional to the product of $y$ and $(L - y)$ "	$\frac{d y}{d t} = k y (L - y)$
"rate of decrease is proportional to $y$ "	$\frac{d y}{d t} = - k y$

Exam Tip

The word proportional is your signal to write $= k \cdot (something)$ . The word decreases or loss signals a negative sign. Always define $k > 0$ and handle sign separately.

Example

Setting up a DE: Newton's Law of Cooling

A cup of coffee cools at a rate proportional to the difference between its temperature $T$ and the room temperature of $20° C$ . Write a DE to model this situation.

Step 1: The quantity changing is temperature $T$ , with respect to time $t$ .

Step 2: "Rate of change" $\Rightarrow \frac{d T}{d t}$

Step 3: "Proportional to the difference between $T$ and $20$ " $\Rightarrow k (T - 20)$

Step 4: The coffee is cooling, so the rate is negative:
$\frac{d T}{d t} = - k (T - 20), k > 0$

If the initial temperature is $90° C$ : initial condition is $T (0) = 90$ .

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Words

DE Form

"rate of change is proportional to

y

\frac{d y}{d t} = k y

"rate of change is proportional to

(y - A)

\frac{d y}{d t} = k (y - A)

"rate of change is proportional to the product of

y

and

(L - y)

\frac{d y}{d t} = k y (L - y)

"rate of decrease is proportional to

y

\frac{d y}{d t} = - k y

AHL 5.14—Setting up a DE, solve by separating variables

What is a Differential Equation?

Setting Up a Differential Equation

9 more sections in this topic

AHL 5.14—Setting up a DE, solve by separating variables

What is a Differential Equation?

Setting Up a Differential Equation

9 more sections in this topic