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., \(f^\prime\text{,}\) \(\frac{dy}{dx}\)) 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 your Understanding

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

\(\frac{dy}{dx} + 1 = y\)
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. In this textbook, what does the abbreviation "DE" stand for?

In this textbook, what does the abbreviation "DE" stand for?

An Ordinary Differential Equation
Correct! In this book, DE is shorthand for Differential Equation.
An Partial Differential Equation
Incorrect! Please review the note “Convention: DE means ODE”.
Dependent Equation
Incorrect. While DE could theoretically stand for Dependent Equation, in this book it always refers to Differential Equation.
Derivative Equation
Incorrect. While DE could theoretically stand for Derivative Equation, is not a standard term in mathematics. In this book it always refers to Differential Equation.

5. What distinguishes an ordinary differential equation (ODE) from a partial differential equation (PDE)?

What distinguishes an ordinary differential equation (ODE) from a partial differential equation (PDE)?

The number of variables the unknown function depends on.
Correct! An ODE has derivatives with respect to a single variable, while a PDE involves multiple variables.
The number of derivatives in the equation.
Incorrect. Please review the definition of ODEs and PDEs.
The number of solutions the equation has.
Incorrect. Please review the definition of ODEs and PDEs.
The number of hours it takes to solve the equation.
Incorrect. Please review the definition of ODEs and PDEs.

6. 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.

7. What notation will this textbook primarily use for derivatives?

What notation will this textbook primarily use for derivatives?

Both prime and Leibniz notation.
Correct! The textbook will use both prime and Leibniz notation for derivatives.
Only prime notation.
Incorrect. While prime notation will be used, Leibniz notation will also be utilized.
Only Leibniz notation.
Incorrect. The book will use both Leibniz and prime notation for derivatives.
Subscript notation.
Incorrect. Subscript notation is not used for derivatives in this textbook.

8. 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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