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., \(f^\prime\text{,}\) \(\ds\frac{dy}{dx}\)) and an equality sign ("="). This distinction helps us identify the following expressions as differential equations:

\begin{equation*} \frac{dy}{dx} + 1 = y, \qquad f^{\prime\prime} + x^2 + 3x = 19, \qquad e^t = \tan(y^\prime) \end{equation*}

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

\begin{equation*} \frac{d^2 y}{dx^2} + 2\frac{dy}{dx}, \qquad x^2 + 3x = 19, \qquad \sin y + e^x = 0 \end{equation*}

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	\(y^\prime\)	\(y^{\prime\prime}\)	\(y^{\prime\prime\prime}\)	\(y^{(4)}\)	\(...\)	\(y^{(n)}\)
Leibniz	\(\ds\frac{dy}{dx}\)	\(\ds\frac{d^2y}{dx^2}\)	\(\ds\frac{d^3y}{dx^3}\)	\(\ds\frac{d^4y}{dx^4}\)	\(...\)	\(\ds\frac{d^ny}{dx^n}\)

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

\(\ds\frac{dy}{dx} + 1 = y\)
Correct! This equation involves a derivative, making it a differential equation.
\(x^2 + 3x = 19\)
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

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

Hint.

There are only 5 Differential Equations in this set.

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