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Section 1.2 Differential Equation Definition
📝: 🎧 Listen.
This textbook introduces a new type of equation that incorporates not only an unknown function but also how that function changes over time. These equations are called
differential equations . In this section, we will provide a clear definition and review the various ways of representing derivatives.
Subsection Definition
📙 Definition 2 .
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, it’s called an
ordinary differential equation (ODE) . If the function depends on more than one variable, it’s called a
partial differential equation (PDE) .
📝: DE \(\Rightarrow\) ODE.
📌 Since this course only covers
ordinary differential equations , any time we say
differential equation or
DE , we mean
ODE .
So, for an equation to qualify as a differential equation, it must include two things: First, it requires a derivative, such as
\(f'\) or
\(\frac{dy}{dx}\text{.}\) Second, it must have an equal sign.
This means all of the following are 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*}
On the other hand, these are not differential equations, either because they do not contain a derivative or because they do not include an equals sign:
\begin{equation*}
\frac{d^2 y}{dx^2} + 2\frac{dy}{dx}, \qquad \sin y + e^x = 0.
\end{equation*}
Checkpoint 3 .
(a) 📖❓ Not Required for a DE.
Which of the following is NOT required for an equation to be classified as a differential equation?
A \(y\) -variable
Correct! A \(y\) -variable is not a requirement for a differential equation.
An unknown function
Incorrect. A differential equation includes an unknown function that we are solving for.
One or more derivatives of an unknown function
Incorrect. The presence of at least one derivative is essential to define a differential equation.
An equals sign, "="
Incorrect. An equals sign is required for an equation to be classified as a differential equation.
(b) 📖❓ Identifying the Differential Equation.
Which of the following equations is a differential equation?
\(\quad y'' + 1 = y\)
Correct! This equation involves a derivative, making it a differential equation.
\(\quad x^2 + 3x = 19\)
Incorrect. This equation contains no derivatives; therefore, it is not a differential equation.
\(\quad \sin y + e^x = 0\)
Incorrect. This equation contains no derivatives; therefore, it is not a differential equation.
\(\quad y^2 + 5 = 0\)
Incorrect. This equation contains no derivatives; therefore, it is not a differential equation.
Subsection Common Derivative Notations
📝: 🎧 Listen.
In this text, we will use several common notations for derivatives. These include prime notation, such as
\(y'\) or
\(y''\text{;}\) Leibniz notation, written as
\(\frac{dy}{dx}\text{;}\) and dot notation, typically reserved for derivatives with respect to time.
Table 4. Common Derivative Notations. Be careful not to mistake \(y^{(4)}\) for \(y\) raised to the fourth power.
\(\textbf{Derivative Order}\)
\(\textbf{Notation}\)
\(1\) st
\(2\) nd
\(3\) rd
\(4\) th
\(n\) th
Prime
\(y'\) \(y''\) \(y'''\) \(y^{(4)}\) \(...\) \(y^{(n)}\)
Leibniz
\(\dfrac{dy}{dx}\) \(\dfrac{d^2y}{dx^2}\) \(\dfrac{d^3y}{dx^3}\) \(\dfrac{d^4y}{dx^4}\) \(...\) \(\dfrac{d^ny}{dx^n}\)
Dot
\(\dot{y}\) \(\ddot{y}\) \(\dddot{y}\) \(\ddddot{y}\) \(...\) \(\text{---}\)
Checkpoint 5 .
The expression
\(z^{(18)}\) is the same as
\(z\) to the power of 18.
You have attempted
of
activities on this page.
