What is a differential equation?

An unknown function tied to one of its own derivatives.

General solution or particular solution?

General: it carries an arbitrary constant. Particular: the constant is fixed by an extra condition, such as a starting value.

When does separation of variables apply?

Only when the right-hand side splits into two factors, one depending on x alone and the other on y alone.

What is exponential decay?

A first-order differential equation where flipping the sign of the constant turns growth into decay.

About 5,730 years, according to OpenStax Calculus Volume 2.

Differential Equations: What They Are and How They Work

In 30 seconds quick read

A differential equation ties an unknown function to at least one of its own derivatives. Its order is set by whichever derivative in the equation is the highest one present: only a first derivative means a first-order equation. Solving one doesn't produce a single number but a whole family of functions, carrying an arbitrary constant that gets pinned down once some extra piece of information about the problem is known. Equations like this describe quantities that change over time: a falling object, a growing population, a decaying radioactive element, a balance that earns interest.

Key Points

An unknown function paired with one of its own derivatives is what makes an equation differential rather than algebraic.
The order comes from the highest derivative written in the equation, nothing else.
A general solution carries an arbitrary constant; it becomes a particular solution once that constant is fixed by a condition, such as a starting value.
A higher order brings more constants into the general solution, matching the number of independent solutions that order allows.
Separation of variables only works when the right-hand side splits into one factor depending on x and another depending on y.
A growing population and a decaying radioactive sample follow the same equation; only the sign of the constant tells them apart.

Key figures

5,730 years The half-life of carbon-14, reported by OpenStax Calculus Volume 2. Source: OpenStax, Calculus Volume 2

Deep Dive

Unlike ordinary equations, the unknown here is a function, tied by the equation to one of its own derivatives. It belongs to the branch of mathematics known as calculus, and it describes phenomena that change over time.

Order and the general form

The order of a differential equation is set by the highest derivative that appears in it. An equation containing only a first derivative is first-order; one reaching a second derivative is second-order, and so on.

According to Wolfram MathWorld, the general form of an ordinary differential equation of order n is written F(x, y, y’, …, y^(n)) = 0, where the unknown function y and its derivatives up to order n appear together. “Ordinary” sets it apart from partial differential equations, which involve more than one independent variable and sit outside this Recap.

Solving a differential equation leads to the general solution: an entire family of functions carrying an arbitrary constant. An ordinary equation of order n has, in general, n solutions independent of each other, which is why it carries that many constants in its general solution. A particular solution comes from fixing that constant with extra information about the problem, such as a starting value.

How to solve them: separation of variables

A differential equation is called separable when it can be written as a function of x alone multiplied by a function of y alone. In that case, separation of variables applies: every y-term moves to one side of the equation, every x-term to the other, and then both sides get integrated on their own.

Worked example: take the equation y’ = (x²-4)(3y+2). The right-hand side is already a product of a function of x alone, (x²-4), and a function of y alone, (3y+2), so the equation is separable. Moving the y-terms to one side and the x-terms to the other, then integrating each side, produces the general solution y = (-2 + Ce^(x³-12x)) / 3, with C an arbitrary constant: changing C picks out a different function from within that same family of solutions.

Not every differential equation lends itself to this treatment: the method depends on that exact product structure, and equations without it call for other tools entirely. The Bernoulli differential equation, for instance, isn’t separable, but a specific substitution turns it into a solvable linear equation.

Where they show up: physics, biology, economics

In physics, the motion of a falling object is written as a differential equation in which the derivative of velocity with respect to time equals gravitational acceleration. In biology, a basic population-growth model starts from the assumption that the growth rate is proportional to the population’s own size; that assumption leads straight to a first-order differential equation. Economics follows the same logic for continuous compound interest: an account balance grows at a rate proportional to the balance itself.

Exponential growth or decay is, in all three settings, a first-order differential equation where the derivative of a quantity is proportional to the quantity itself. Only the sign of the constant changes: positive for growth, negative for decay. Solving this kind of equation naturally produces an exponential function.

Radioactive decay and Newton’s law of cooling

For exponential decay, half-life is the time a quantity takes to drop by half. According to OpenStax, carbon-14 has a half-life of about 5,730 years — the textbook example of radioactive decay modeled with a differential equation, and the reason radiocarbon dating works at all.

Newton’s law of cooling models an object cooling down with a differential equation in which the cooling rate is proportional to the difference between the object’s temperature and that of its surroundings.

The four relationships covered so far, side by side:

Phenomenon	What the equation ties together	Sign of the constant
Population growth	derivative of the quantity, the quantity itself	positive
Radioactive decay (carbon-14)	derivative of the quantity, the quantity itself	negative
Newton’s cooling	cooling rate, temperature difference	negative
Continuous compound interest	derivative of the balance, the balance itself	positive

Slide deck

Slides ready to download and make your own in PowerPoint or Google Slides, with speaker notes. Pick the Flash cut or the Full one.

Slide 1 of the presentation on Differential Equations: Differential Equations

Slide 2 of the presentation on Differential Equations: A multiplying bacterium and an interest-bearing account obey the same equation

Slide 3 of the presentation on Differential Equations: From definition to real models

Slide 4 of the presentation on Differential Equations: Chapter 01: What a differential equation is

Slide 5 of the presentation on Differential Equations: The anatomy of a differential equation

Slide 6 of the presentation on Differential Equations: Chapter 02: The method: separation of variables

Slide 7 of the presentation on Differential Equations: Isolate, then integrate

Slide 8 of the presentation on Differential Equations: Y equals, in parentheses, negative two plus C times e to the power of x cubed minus 12x, close parentheses, all over 3: where does this solution come from?

Slide 9 of the presentation on Differential Equations: Chapter 03: Where it shows up

Slide 10 of the presentation on Differential Equations: A falling object · Population · Compound interest

Slide 11 of the presentation on Differential Equations: Chapter 04: Growth and decay

Slide 12 of the presentation on Differential Equations: Growth versus decay

Slide 13 of the presentation on Differential Equations: Decay and cooling, same model

Slide 14 of the presentation on Differential Equations: Differential equations aren't closed theory from a calculus textbook

Slide 15 of the presentation on Differential Equations: Which of these has a derivative proportional to the quantity itself?

Slide 16 of the presentation on Differential Equations: Now, review time

Flash10 slidesThe essential thread, to present in class Full16 slidesEvery chapter and the deeper detail

Common myths

✗ Myth A common assumption is that every differential equation can be cracked with a formula, the way an algebraic equation can.

✓ Reality Separation of variables only works under one specific condition: the equation has to break down into a piece that depends solely on x and another that depends solely on y. Outside that case, the technique simply doesn't apply, and other tools are needed.
✗ Myth Differential equations can look like a closed chapter from a calculus textbook, with no real-world use.

✓ Reality OpenStax shows the same equation describing a falling object, a growing population, and a bank balance earning continuous interest — three unrelated settings, one shared structure.
✗ Myth Solving a differential equation is assumed to land on one exact answer, the way an algebraic equation does.

✓ Reality What comes out is a general solution with an open constant; it only narrows to one particular answer once an extra condition, such as a starting value, pins that constant down.

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Grade 0/10 0/5

1 What does a differential equation contain, by definition?

2 What is the order of a differential equation?

3 When can separation of variables be used?

4 True or false: the same equation describes both population growth and radioactive decay.

5 According to Wolfram MathWorld, how many independent solutions does an ordinary differential equation of order n generally have?

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Frequently asked questions

What's the difference between a regular equation and a differential equation?

A regular equation looks for a number that satisfies it; a differential equation looks for a function, and it includes at least one of that function's derivatives.

What is the order of a differential equation?

Check which derivative is the highest one written in the equation — that's its order.

When can separation of variables be used?

First check whether the equation splits into a piece involving only x and a piece involving only y. If it does, gather every y-term on one side and every x-term on the other, then integrate each side on its own.

What are differential equations actually used for?

They model a falling object in physics, a growing population in biology, the decay of carbon-14, an object cooling down, and a balance growing under continuous compound interest.