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!
Reading Questions Check-Point Questions
1. Differential equations differ from standard equations in that they contain \(\ul{\qquad}\).
Differential equations differ from standard equations in that they contain \(\fillinmath{XXX}\)
- solutions
Incorrect. While this statement is generally true, it is not what makes it different from any other equation.
-
\(y\) variables
Incorrect. Any equation could contain a \(y\) variable.
- unknowns
Incorrect. Most equations contain an unknown.
- derivatives
Correct! If an equation contains a derivative, it is a differential equation.
2. Which of the following best describes a differential equation?
Which of the following best describes a differential equation?
- An equation involving only algebraic expressions.
Incorrect. A differential equation involves derivatives.
- An equation involving functions and their derivatives.
Correct! A differential equation involves functions and their derivatives.
- An equation involving trigonometric functions.
Incorrect. A differential equation could involve trigonometric functions, but it is not a defining characteristic.
- An equation that changes over time.
Incorrect. A differential equation could describe a system that changes over time, but it is not a defining characteristic.
3. What distinguishes a differential equation from a standard equation?
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