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

## Section1.1An 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!

#### 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.

#### 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.
• An equation involving functions and their derivatives.
• An equation involving trigonometric functions.
• An equation that changes over time.

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

What distinguishes a differential equation from a standard equation?
• It contains an unknown variable.
• Incorrect. Both standard and differential equations contain unknown variables.
• It contains a derivative.
• Correct! A differential equation contains one or more derivatives, which differentiates it from a standard equation.
• It contains a $$y$$ variable.
• Incorrect. Any equation could contain a $$y$$ variable.