What Is a Differential Equation?

A differential equation is an equation that involves derivatives or differentials of an unknown function. If the unknown function depends on a single independent variable, the equation is an ordinary differential equation; if it depends on two or more, partial derivatives appear and the equation is a partial differential equation.

Quick Reference

Term	Meaning
Differential equation	An equation involving derivatives or differentials of an unknown function
Ordinary differential equation (ODE)	One independent variable, so only ordinary derivatives appear
Partial differential equation (PDE)	Two or more independent variables, so partial derivatives appear
Dependent variable	The unknown function being solved for, usually $y$ , $u$ , or $V$
Independent variable	The variable the unknown function depends on, usually $x$ or $t$
y', y'', y''', y^{(4)}, \dots, y^{(n)}	First, second, third, fourth, and $n$ th derivatives of $y$
$\dot{y}, \ddot{y}$	$d y / d t$ and $d^{2} y / d t^{2}$ , dot notation used when the variable is time

The Definition

A differential equation, as the words suggest, is an equation that involves derivatives or differentials of an unknown function.

A differential equation is an equation that relates an unknown function to one or more of its derivatives.

The following are all examples of differential equations:

\frac{d^{2} y}{d x^{2}} + 5 y = 0

{(x \frac{d y}{d x} - y)}^{2} = (x + y)^{2}

\left\{ \begin{aligned} &\frac{dy}{dt} - 3\frac{dx}{dt} = 0 \\ &\frac{dy}{dt} + 5\frac{dx}{dt} = e^{-t} \end{aligned} \right.

\frac{\frac{d y}{d x}}{{[1 + {(\frac{d y}{d x})}^{2}]}^{3 / 2}} = 1

x^{2} y d x + x^{3} y^{2} d y = 0

x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = z

Notice how varied these look. Some are written with $d^{2} y / d x^{2}$ , some with differentials $d x$ and $d y$ , one is a system of two equations, and the last uses partial derivatives. What unites them is that in each case the unknown is a function, and the equation constrains that function through its derivatives.

Ordinary vs. Partial Differential Equations

Equations that involve a single independent variable, so that only ordinary derivatives occur, are known as ordinary differential equations (ODEs). When there is more than one independent variable, partial derivatives are involved and the equation is called a partial differential equation (PDE).

The test is simple: count the independent variables. If the unknown function is $y (x)$ , a function of $x$ alone, every derivative in the equation is an ordinary derivative and the equation is an ODE. If the unknown is $u (x, t)$ , derivatives with respect to $x$ hold $t$ fixed and vice versa, so they are partial derivatives and the equation is a PDE.

This book is about ordinary differential equations. Partial differential equations appear here only so that you can recognize them.

Notation for Derivatives

The expressions y', y'', y''', y^{(4)}, \dots, y^{(n)} represent the first, second, third, fourth, and $n$ th derivatives of $y$ with respect to the independent variable under consideration.

For example, y''' means $d^{3} y / d x^{3}$ when the independent variable is $x$ . Note that once you reach the fourth derivative, the prime notation becomes hard to read, so a parenthesized superscript is used instead: $y^{(4)}$ , not y''''. The parentheses matter, because $y^{4}$ without them would mean the fourth power of $y$ .

In physics, when the independent variable is time $t$ , dots are sometimes used instead of primes. Therefore $\dot{y} = d y / d t$ and $\ddot{y} = d^{2} y / d t^{2}$ . This convention goes back to Newton and is still standard in mechanics.

Worked Examples

Classify each of the following as an ODE or a PDE, and identify the unknown function and the independent variables.

$\frac{d y}{d x} = 5 x + 4 y$ , also written y' = 5x + 4y
$y + {(\frac{d^{3} y}{d x^{3}})}^{2} = \sin x$ , also written y + (y''')^2 = \sin x
$\frac{\partial^{2} V}{\partial x^{2}} + \frac{\partial^{2} V}{\partial y^{2}} = f (x, y)$ , where $f$ is a given function
$\frac{\partial u}{\partial t} = \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}}$
$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = ν \frac{\partial^{2} u}{\partial x^{2}}$ , where $ν$ is a constant

Solution

ODE. The unknown function is $y$ , regarded as a function of the single variable $x$ . Only the ordinary derivative $d y / d x$ appears.
ODE. Again $y$ is a function of $x$ alone. The third derivative is squared, but that affects the degree of the equation, not whether it is ordinary.
PDE. The unknown $V$ is a function of the two variables $x$ and $y$ , so the second derivatives are partial. This is Poisson's equation.
PDE. The unknown $u$ is a function of $x$ , $y$ , and $t$ . This is the two-dimensional heat equation.
PDE. The unknown $u$ is a function of $x$ and $t$ . This is Burgers' equation, which combines the transport term $u \partial u / \partial x$ with the diffusion term $ν \partial^{2} u / \partial x^{2}$ .

Verify that $y = e^{3 x}$ is a solution of y' - 3y = 0, and that $y = e^{2 x}$ is not.

Solution

For $y = e^{3 x}$ we have y' = 3e^{3x}, so

y' - 3y = 3e^{3x} - 3e^{3x} = 0 .

The equation reduces to $0 = 0$ for every real $x$ , so $y = e^{3 x}$ is a solution.

For $y = e^{2 x}$ we have y' = 2e^{2x}, so

y' - 3y = 2e^{2x} - 3e^{2x} = -e^{2x},

which is never zero. So $y = e^{2 x}$ is not a solution.

Checking a proposed solution is always this mechanical: substitute the function and its derivatives into the equation and see whether both sides agree identically.

Exercises

State whether each equation is an ODE or a PDE.

$\frac{d^{2} y}{d t^{2}} + 9 y = \cos t$
$\frac{\partial w}{\partial s} = 4 \frac{\partial^{2} w}{\partial r^{2}}$
${(\frac{d y}{d x})}^{3} = x y$

Solution

ODE. The unknown $y$ depends only on $t$ .
PDE. The unknown $w$ depends on both $r$ and $s$ .
ODE. The unknown $y$ depends only on $x$ . The cube raises the degree but not the number of independent variables.

Show that $y = c e^{- x^{2}}$ is a solution of y' + 2xy = 0 for every constant $c$ .

Solution

Differentiating with the chain rule, y' = c e^{-x^2} \cdot (-2x) = -2xc e^{-x^2}. Substituting,

y' + 2xy = -2xc e^{-x^2} + 2x \cdot c e^{-x^2} = 0 .

The identity holds for every real $x$ and every value of $c$ , so the whole one-parameter family is made of solutions.

Frequently Asked Questions

What exactly is a differential equation?

It is an equation that involves derivatives or differentials of an unknown function. The unknown is a function, not a number, and the equation says something about how that function and its rates of change are related. For instance, y' = 2y says: find every function that equals half of its own derivative.

How do I tell an ODE from a PDE at a glance?

Look for the partial derivative symbol $\partial$ . If every derivative is written with a straight $d$ , as in $d y / d x$ or $d^{3} y / d x^{3}$ , the unknown depends on one variable and the equation is ordinary. If you see $\partial u / \partial x$ or $\partial^{2} u / \partial t^{2}$ , the unknown depends on more than one variable and the equation is partial.

Why is

y^{(4)}

written with parentheses instead of y''''?

Four tick marks are hard to count at a glance, so the parenthesized superscript takes over from the fourth derivative on. The parentheses are not optional: $y^{(4)}$ means the fourth derivative, while $y^{4}$ means $y$ raised to the fourth power. Dropping them changes the meaning of the equation entirely.

Does a differential equation have to be solved for the highest derivative?

No. An equation such as \left(x y' - y\right)^2 = (x+y)^2 is a perfectly good differential equation even though y' is tangled inside a square. Many solution techniques do assume the equation has been solved for the highest derivative, which is why the general first-order form F(x,y,y')=0 is often replaced by y' = f(x,y) when that step is possible.

Can a differential equation involve differentials like

d x

and

d y

instead of derivatives?

Yes. The equation $x^{2} y d x + x^{3} y^{2} d y = 0$ is written in differential form. Dividing through by $d x$ recovers the familiar derivative form $x^{2} y + x^{3} y^{2} \frac{d y}{d x} = 0$ . Differential form is especially convenient for exact equations and separable equations, where it treats $x$ and $y$ symmetrically.

Is a system of differential equations still a differential equation?

It is a system, and the pair

\frac{d y}{d t} - 3 \frac{d x}{d t} = 0, \frac{d y}{d t} + 5 \frac{d x}{d t} = e^{- t}

must be solved for both unknown functions $x (t)$ and $y (t)$ at once. Systems get their own treatment in the chapter on systems of differential equations, where matrix methods make them tractable.

Quick Reference

The Definition

Ordinary vs. Partial Differential Equations

Notation for Derivatives

Worked Examples

Exercises

Frequently Asked Questions

Quick Links

Social Media