##
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).

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*}

###
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.

You have attempted

of

activities on this page.