# Interactive Differential Equations: A Step-by-Step Approach to Methods & Modeling

## Section1.2Definition

Here is the formal definition of a differential equation.

### Definition1.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).

### Note2.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*}

### Note3.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!

#### 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
• 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.

Hint.
There are only 5 Differential Equations in this set.