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Section 1.1 An Analogy

When you’re learning something new, it’s often helpful to connect it to concepts you already know. To grasp what a differential equation is, let’s first compare it to standard equations that might feel more familiar. Consider the following three equations, where we aim to solve for \(y\text{:}\)

\begin{equation*}
y + 3 = 11
\end{equation*}

\begin{equation*}
y + 3x = 11
\end{equation*}

\begin{equation*}
y^\prime + 3x = 11
\end{equation*}

All three are equations with the same goal—finding the unknown \(y\text{.}\) However, only the third equation is a *differential equation* because it contains a derivative.

Now, let’s try solving for \(y\) in each case.

\begin{align*}
y + 3 =\amp\ 11\\
y =\amp\ 8
\end{align*}

\begin{align*}
y + 3x =\amp\ 11\\
y =\amp\ 11 - 3x
\end{align*}

\begin{align*}
y^\prime + 3x =\amp\ 11\\
y^\prime =\amp\ 11 - 3x\\
y =\amp\ ?
\end{align*}

In the first equation, we found that \(y\) is a number, and in the second, it’s a function of \(x\text{.}\) But in the third equation, how do we solve for \(y\) when there is a derivative attached to it? This is exactly the kind of question that differential equations aim to answer.

We’ll dive deeper into solving these types of equations soon. For now, there’s still plenty more to learn about the basics, so let’s keep going!

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Reading Questions Check your Understanding

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1. *Differential equations differ from standard equations in that they have *.

*Differential equations differ from standard equations in that they have *

- a solution
Incorrect. While this statement is generally true, it is not what makes it different from any other equation.

- a \(y\) variable
Incorrect. Any equation could contain a \(y\) variable.

- an unknown
Incorrect. Most equations contain an unknown.

- a derivative
Correct! If an equation contains a derivative, it is a differential equation.

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2. *Which of the following best describes a differential equation?*

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3. *What distinguishes a differential equation from a standard equation?*

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