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Section 1.1 Connection to Algebra & Calculus
📝: 🎧 Listen.
To learn something new, it helps to connect it to something you already know. Let’s begin by linking differential equations to algebra.
Suppose you’re asked to solve each of the equations for \(y\text{:}\)
\begin{equation*}
y + 2 = 11,\quad y + 2t = 11,\quad y^3 + 2t = 11.
\end{equation*}
The goal of algebra is to determine what goes in place of \(y\) to make both sides equal. Instead of guessing, you learned rules for isolating \(y\text{.}\) Applying those rules, we find:
\begin{align*}
y = 9,\quad y = 11 - 2t,\quad y^3 \amp = 11 - 2t \\
y \amp = \left(11 - 2t\right)^{1/3}
\end{align*}
Now suppose you’re asked to solve a new equation:
\begin{equation*}
y' + 2t = 11.
\end{equation*}
As before, we isolate \(y'\text{:}\)
\begin{equation*}
y' = 11 - 2t.
\end{equation*}
So what function has a derivative equal to \(11 - 2t\text{?}\) In other words, what goes in place of \(y\) to make this equation true?
From calculus, you might recognize this as an antiderivative problem. So the
\(y\) we seek is
\(y = 11t - t^2 + C\text{.}\)
Finding this
\(y\) is exactly what solving a differential equation aims to do. Most problems won’t be quite that straightforward, but the goal is the same: find a function that fits in place of
\(y\) and makes the equation true.
Checkpoint 1 .
(a) 📖❓ Solving for an Unknown.
How are differential equations related to equations from algebra?
You are solving an equation for an unknown.
Correct! When you solve a differential equation, the unknown is a function rather than a number.
You are only evaluating derivatives at a single point.
A differential equation is not about evaluating a derivative at a point, but about finding a function that satisfies an equation.
You are checking whether a given function is differentiable.
Differentiability may be relevant, but the primary objective is to determine the unknown function.
You are simplifying an expression without solving for a variable.
Simplification alone is not sufficient; solving a differential equation means finding the function that replaces the unknown.
(b) 📖❓ Role of Derivatives.
How are differential equations related to calculus?
Differential equations are equations containing derivatives of the unknown function.
Correct! Differential equations rely on derivatives, which are a central topic in calculus.
Differential equations contain integrals.
While integrals may appear when solving them, differential equations are defined by the presence of derivatives, not integrals.
Differential equations find derivatives of a function.
Actually, differential equations involve finding the function itself, not just its derivatives.
The limit of a differential equation is an algebraic equation.
Please reread this section.
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