π Example 312. Evaluating a Function.
\begin{equation*}
f(4) = 2(4) + 3 = 11.
\end{equation*}
This tells us that when the input is 4, the function outputs 11.
Section Function Notation
Function notation is a cornerstone of modern mathematics and science. It allows us to describe relationships between variables in a compact, precise, and flexible way. While the concept may seem familiar, mastering its nuances is essential for understanding differential equations and the systems they model.
A function is a rule that assigns exactly one output value to each input. For example, \(f(x) = x^2\) takes an input \(x\text{,}\) squares it, and returns the result. The notation \(f(x)\) not only names the function but also reminds us which variable it depends on.
In the context of differential equations, function notation frequently appears with derivativesβexpressions like \(f'(x)\) or \(\frac{dy}{dx}\) describe how a quantity changes. These are the building blocks for modeling motion, population growth, heat flow, electrical current, and countless other processes.
Function notation also makes it easy to represent more complex relationships. For instance, in the differential equation \(\frac{dy}{dx} = f(x) \cdot g(y)\text{,}\) we describe how the rate of change of \(y\) depends both on the independent variable \(x\) and the dependent variable \(y\) through two separate functions.
π Example 313. Function Composition.
Consider \(f(x) = \sin(x)\) and \(g(y) = e^y\text{.}\) The composition \(g(f(x))\) means we evaluate \(f(x)\) first, then plug the result into \(g\text{:}\)
\begin{equation*}
g(f(x)) = e^{\sin(x)}.
\end{equation*}
This compact notation is especially useful in modeling layered or dependent processes.
Initial conditions often appear alongside function notation in differential equations. For example, if \(\frac{dy}{dx} = 3x\) and \(y(0) = 5\text{,}\) the function \(y(x)\) must satisfy both the differential equation and this initial value.
π Example 314. Function Notation in Initial Value Problems.
Solve the initial value problem:
\begin{equation*}
\frac{dy}{dx} = 2x, \qquad y(1) = 4.
\end{equation*}
First, find the general solution:
\begin{equation*}
y(x) = x^2 + C.
\end{equation*}
Now use the initial condition:
\begin{equation*}
4 = 1^2 + C \quad \Rightarrow \quad C = 3.
\end{equation*}
So the particular solution is:
\begin{equation*}
y(x) = x^2 + 3.
\end{equation*}
Function notation also plays a central role in the structure of higher-order differential equations, where derivatives and coefficients are expressed using functional relationships.
Consider the general form:
\begin{equation*}
\frac{d^n y}{dt^n} + a(t)y = k(t).
\end{equation*}
This describes how a function \(y(t)\) evolves over time, incorporating both its derivatives and time-dependent coefficients. Letβs break down the components.
Key Components of Function Notation.
1. The Function: \(y = y(t)\) indicates that \(y\) is a function of time \(t\text{.}\) For example, \(y(t)\) might represent the temperature at time \(t\text{,}\) or the position of an object in motion.
2. Derivatives: The term \(\frac{d^n y}{dt^n}\) refers to the \(n\)th derivative of \(y\) with respect to \(t\text{.}\) Derivatives describe how a quantity changes. For example:
-
\(\frac{dy}{dt}\) is the rate of change (velocity).
-
\(\frac{d^2y}{dt^2}\) is the second derivative (acceleration).
3. Coefficients and Forcing Terms: The functions \(a(t)\) and \(k(t)\) modify the behavior of the system. For example:
-
\(a(t)y\) scales the function \(y(t)\) based on time.
-
\(k(t)\) represents an external input or forcing term.
Interpreting an Equation.
Letβs analyze this second-order differential equation:
\begin{equation*}
\frac{d^2y}{dt^2} + 5\frac{dy}{dt} + 6y = 10e^{-t}.
\end{equation*}
-
\(\frac{d^2y}{dt^2}\) is the second derivativeβhow fast the rate of change is changing.
-
\(5\frac{dy}{dt}\) is a damping term proportional to velocity.
-
\(6y\) scales the function itself.
-
\(10e^{-t}\) is a forcing term that decays over time.
These components together describe a system with internal resistance and an external influence.
When interpreting a differential equation, ask:
-
What is the order of the highest derivative?
-
Are the coefficients constant or variable?
-
Is there a forcing term (like \(k(t)\)) driving the system?
These questions help you understand the systemβs structure and behavior.
π Example 315. Breaking Down an Equation.
Consider:
\begin{equation*}
\frac{dy}{dx} = 3x^2y.
\end{equation*}
This is a first-order differential equation. The rate of change of \(y\) depends on both \(x\) and \(y\text{.}\) We can separate the variables:
\begin{equation*}
\frac{1}{y} \, dy = 3x^2 \, dx.
\end{equation*}
Recognizing this structure makes it easier to solve.
In summary, function notation is more than a way to write formulasβitβs a powerful language for describing change. It allows us to interpret, analyze, and solve differential equations across a wide range of applications.
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