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Section 1.4 Terms & Coefficients
Before we begin solving differential equations, we need a shared vocabulary for discussing their structure. In this section, we introduce the concepts of “terms” and “coefficients”, which help us describe what an equation is composed of and prepare us to classify and solve different types of equations.
Subsection Terms
In mathematics, the word “term” usually refers to a part of an expression separated by plus or minus signs. The same idea applies to differential equations.
So, a
term in a
differential equation is any piece of the equation separated by addition or subtraction. For example, we might have something like:
\begin{equation*}
\text{term}\ 1 + \text{term}\ 2 - \text{term}\ 3 = \text{term}\ 4 - \text{term}\ 5.
\end{equation*}
This example has three terms on the left-hand side and two on the right.
Terms in a differential equation can include the
dependent variable , its derivatives, constants, or other functions of the
independent variable . If a term does not involve the dependent variable or its derivatives, we call it a
free term .
For example, consider the five-term differential equation with dependent variable \(y\text{:}\)
\begin{equation*}
\dfrac{3}{x} y^{(6)} + 5.3 y'' + x^2 y' + y = \dfrac12\ln(x)\text{.}
\end{equation*}
It is common to group terms by the different forms of the dependent variable:
\(y^{(6)}\ \text{term}\) \(y''\ \text{term}\) \(y'\ \text{term}\) \(y\ \text{term}\) \(\text{free term}\)
\(\dfrac{3}{x} y^{(6)}\) \(5.3 y''\) \(x^2 y'\) \(y\) \(\dfrac12\ln(x)\)
📝: “Free” terms.
📌 Think of a
free term as being “free” of the dependent variable.
📌 These are often called
constant terms , but this can be misleading, as they may contain independent variables.
Every differential equation has a
free term , even if it is not written explicitly. For example,
\(y' = -3y\) can be rewritten as
\(y' + 3y = 0\text{,}\) with
\(0\) as the free term.
Checkpoint 9 .
Identify the free term in the differential equation
\begin{equation*}
3t^2 y' - 4t^2 = \dfrac{1}{t} y\text{.}
\end{equation*}
Subsection Coefficients
In a differential equation, a
coefficient is a constant or a function of the independent variable that multiplies the dependent variable or one of its derivatives. Coefficients determine the equation’s structure and influence which solution methods are applicable.
For example, in the equation
\begin{equation}
y'' - 3y' + 2y = 0,\tag{1.2}
\end{equation}
the coefficients of \(y''\text{,}\) \(y'\text{,}\) and \(y\) are the constants \(1\text{,}\) \(-3\text{,}\) and \(2\text{,}\) respectively.
Coefficients need not be constants. They can also be functions of the
independent variable . For instance, in the equation
\begin{equation*}
t^2 y'' + 5t y' + 6y = \sin(t),
\end{equation*}
the coefficient of \(y''\) is \(t^2\text{,}\) the coefficient of \(y'\) is \(5t\text{,}\) and the coefficient of \(y\) is \(6\text{.}\)
In general, for something to be considered a
coefficient , it must not involve the dependent variable.
Checkpoint 10 .
(a)
Identify the coefficient of \(y'\) in the differential equation
\begin{equation*}
5y'' + 2\cos(t)y' - y = 7
\end{equation*}
\(\quad \cos(t)\)
Incorrect, \(\cos(t)\) is only part of the coefficient of \(y'\) .
\(\quad 2\cos(t)\)
Correct! \(2\cos(t)\) is the coefficient of the term involving \(y'\) .
\(\quad 2\)
Incorrect, \(2\) is only part of the coefficient of \(y'\) .
\(\quad 7\)
Incorrect. \(7\) is the constant on the right-hand side of the equation.
(b)
Click on each of the coefficients in the differential equation below.
\(t\) \(\dfrac{d^2y}{dt^2}\) \(\ + \ \) \(t^2\) \(y^2\) \(\ - \ \) \(4\) \(y'\) \(\ = \ \) \(y^{-1}\) \(t\) \(\ + \ \) \(\sin(t)\)
\(\vphantom{vertical-space}\)
Hint .
Look for the dependent variable in each term. The coefficient is the constant or function that multiplies the dependent variable.
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