DE-BOOK Definition

Section 1.2 Definition

Here is the formal definition of a differential equation.

Definition 1. Differential Equation.

A differential equation (DE) is an equation that involves one or more derivatives of an unknown function. If the function depends on a single variable, the equation is an ordinary differential equation (ODE). Otherwise, it is called a partial differential equation (PDE).

Note 2. Convention: DE means ODE.

Since this text focuses exclusively on ordinary differential equations, any use of DE will imply ODE.

According to the definition, a differential equation must include at least one derivative (e.g.,

f^{'},

\frac{d y}{d x}

) and an equality sign ("="). This distinction helps us identify the following expressions as differential equations:

\frac{d y}{d x} + 1 = y, f^{''} + x^{2} + 3 x = 19, e^{t} = \tan (y^{'})

In contrast, the following are not differential equations because they either lack a derivative or an equality sign:

\frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x}, x^{2} + 3 x = 19, \sin y + e^{x} = 0

Note 3. Derivative Notation.

We will use either prime notation or Leibniz notation to denote derivatives. For higher-order derivatives, the following conventions apply:

	Derivative Order
	1	2	3	4	$. . .$	n
Prime	$y^{'}$	$y^{''}$	$y^{'''}$	$y^{(4)}$	$. . .$	$y^{(n)}$
Leibniz	$\frac{d y}{d x}$	$\frac{d^{2} y}{d x^{2}}$	$\frac{d^{3} y}{d x^{3}}$	$\frac{d^{4} y}{d x^{4}}$	$. . .$	$\frac{d^{n} y}{d x^{n}}$

Be careful not to confuse $y^{(7)}$ with $y$ raised to the power of 7!

Reading Questions Check-Point Questions

1. An equation that contains an "=" sign and at least one derivative is called a derivative equation.

An equation that contains an "=" sign and at least one derivative is called a derivative equation

True
Incorrect, derivative equation is not a standard term in mathematics.
False
Correct!

2. The expression $z^{(18)}$ is the same as $z$ to the power of 18.

The expression $z^{(18)}$ is the same as $z$ to the power of 18

True
Incorrect. Please read the note on derivative notation.
False
Correct!

3. Identify the differential equation.

Identify the differential equation

$\frac{d y}{d x} + 1 = y$
Correct! This equation involves a derivative, making it a differential equation.
$x^{2} + 3 x = 19$
Incorrect. This equation does not contain any derivatives, so it is not a differential equation.
$\sin y + e^{x} = 0$
Incorrect. This equation does not contain any derivatives, so it is not a differential equation.
$y^{2} + 5 = 0$
Incorrect. This equation does not contain any derivatives, so it is not a differential equation.

4. Which of the following is NOT required for an equation to be classified as a differential equation?

Which of the following is NOT required for an equation to be classified as a differential equation?

An unknown function.
Incorrect. A differential equation does include an unknown function, which we are solving for.
An $x$ -variable.
Correct! An $x$ -variable is not a requirement for a differential equation.
A derivative.
Incorrect. The presence of at least one derivative is essential to define a differential equation.
An "=" sign.
Incorrect. An equality sign is required for an equation to be classified as a differential equation.

5. Click on all the Differential Equations.

Click on all the Differential Equations

$\frac{d y}{d x} + 3 y - 1$	$x^{2} + 2 x - 5 = 0$	$\sin (x) + \cos (x) = 1$
$\frac{d^{2} y}{d x^{2}} - y = e^{x}$	$y + 2 x$	$y = y^{'}$
$\ln (x) + \frac{d y}{d x} = x^{2}$	$\sqrt{x} + 5 = 3 x$	$\frac{d^{3} z}{d t^{3}} - 4 z = \cos (t)$
$x^{2} + y^{2} = r^{2}$	$f^{'} (x) + f (x) = 2$	$\frac{1}{x} + 3$

Hint.

There are only 5 Differential Equations in this set.

You have attempted 1 of 6 activities on this page.

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Section 1.2 Definition

Definition 1. Differential Equation.

Note 2. Convention: DE means ODE.

Note 3. Derivative Notation.

Reading Questions Check-Point Questions

1. An equation that contains an "=" sign and at least one derivative is called a derivative equation.

2. The expression z(18) is the same as z to the power of 18.

3. Identify the differential equation.

4. Which of the following is NOT required for an equation to be classified as a differential equation?

5. Click on all the Differential Equations.

2. The expression $z^{(18)}$ is the same as $z$ to the power of 18.