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Section What is a Differential Equation?
π: π§ Listen.
In this section, weβll explore what a differential equation is, starting with an analogy, and then moving to a formal definition.
Subsection Connection to Algebra and Calculus
To learn something new, it helps to connect it to something you already know. So letβs begin by linking differential equations to algebra.
Suppose youβre asked to solve each of the following equations for
\(y\text{:}\)
\begin{equation*}
y + 2 = 11, \qquad y + 2t = 11, \qquad y^3 + 2t = 11
\end{equation*}
The goal of algebra is to figure out what number or function goes in place of
\(y\) to make both sides equal. Instead of guessing, you learned rules for isolating
\(y\text{.}\) If we apply those rules, we get:
\begin{align*}
y = 9, \qquad y = 11 - 2t, \qquad y^3 \amp = 11 - 2t \\
y \amp = \left(11 - 2t\right)^{1/3}
\end{align*}
Now suppose youβre asked to solve a new equation:
\begin{equation*}
y' + 2t = 11
\end{equation*}
As before, we isolate
\(y'\text{,}\) giving:
\begin{equation*}
y' = 11 - 2t
\end{equation*}
So what function has a derivative of
\(11 - 2t\text{?}\) In other words, what goes in place of
\(y\) to make this equation true?
From calculus, you might recognize this as an anti-derivative problem. So the solution is:
\begin{equation*}
y = \int (11 - 2t)\, dt = 11t - t^2 + C.
\end{equation*}
Believe it or not, you just solved a differential equation using algebra and calculus! Most problems wonβt be quite that straightforward, but the goal is the same: find a function that fits in place of
\(y\) and makes the equation true.
Subsection Definition
π: π§ Listen.
Essentially, a differential equation is any equation where the unknown has a derivative applied to it. Hereβs the formal definition.
π 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, itβs called an
ordinary differential equation (ODE) . If the function depends on more than one variable, itβs called a
partial differential equation (PDE) .
π: DE \(\Rightarrow\) ODE.
π 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 needs a derivativeβsomething like
\(f'\) or
\(\frac{dy}{dx}\text{.}\) Second, it must have an equals sign.
That 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 equationsβeither because they donβt contain a derivative, or because they donβt 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 2 . πβDifferential Equation Definition.
(a) πβ 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 does include an unknown function, which 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 equal sign, "="
Incorrect. An equality sign is required for an equation to be classified as a differential equation.
(b) πβ 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 does not contain any derivatives, so it is not a differential equation.
\(\quad \sin y + e^x = 0\) Incorrect. This equation does not contain any derivatives, so it is not a differential equation.
\(\quad y^2 + 5 = 0\) Incorrect. This equation does not contain any derivatives, so it is not a differential equation.
Subsection Common Derivative Notations
π: π§ Listen.
In this text, weβll use several common notations for derivatives. These include prime notation, like
\(y'\) or
\(y''\text{;}\) Leibniz notation, using
\(\frac{dy}{dx}\text{;}\) and dot notation, which is typically reserved for derivatives with respect to time.
\begin{equation*}
\text{Notation}
\end{equation*}
\begin{equation*}
\text{Derivative Order}
\end{equation*}
\begin{equation*}
1\text{st}
\end{equation*}
\begin{equation*}
2\text{nd}
\end{equation*}
\begin{equation*}
3\text{rd}
\end{equation*}
\begin{equation*}
4\text{th}
\end{equation*}
\begin{equation*}
...
\end{equation*}
\begin{equation*}
n\text{th}
\end{equation*}
\begin{equation*}
\text{Prime}\vphantom{\frac11}
\end{equation*}
\begin{equation*}
y'
\end{equation*}
\begin{equation*}
y''
\end{equation*}
\begin{equation*}
y'''
\end{equation*}
\begin{equation*}
y^{(4)}
\end{equation*}
\begin{equation*}
...
\end{equation*}
\begin{equation*}
y^{(n)}
\end{equation*}
\begin{equation*}
\text{Leibniz}
\end{equation*}
\begin{equation*}
\ds\frac{dy}{dx}
\end{equation*}
\begin{equation*}
\ds\frac{d^2y}{dx^2}
\end{equation*}
\begin{equation*}
\ds\frac{d^3y}{dx^3}
\end{equation*}
\begin{equation*}
\ds\frac{d^4y}{dx^4}
\end{equation*}
\begin{equation*}
...
\end{equation*}
\begin{equation*}
\ds\frac{d^ny}{dx^n}
\end{equation*}
\begin{equation*}
\text{Dot}\vphantom{\frac11}
\end{equation*}
\begin{equation*}
\dot{y}
\end{equation*}
\begin{equation*}
\ddot{y}
\end{equation*}
\begin{equation*}
\dddot{y}
\end{equation*}
\begin{equation*}
\ddddot{y}
\end{equation*}
\begin{equation*}
...
\end{equation*}
\begin{equation*}
\text{---}
\end{equation*}
π: Notation.
π Be careful not to mistake
\(y^{(4)}\) as
\(y\) raised to the power of 4.
π Dot notation is normally reserved for derivatives with respect to time.
Checkpoint 3 . πβ Understanding Derivative Notation.
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!
Subsection π€ Wrap-Up
ποΈ \(\textbf{Key Takeaways...}\)
A differential equation involves an unknown function and at least one of its derivatives.
It must contain an equals sign to be considered an equation.
Differential equations are a natural extension of algebra and calculus.
Notation varies by context but generally falls into prime, Leibniz, or dot notation.
You have attempted
of
activities on this page.
