Skip to main content
Contents Index Calc Dark Mode Prev Next Profile \(\newcommand\DLGray{\color{Gray}}
\newcommand\DLO{\color{BurntOrange}}
\newcommand\DLRa{\color{WildStrawberry}}
\newcommand\DLGa{\color{Green}}
\newcommand\DLGb{\color{PineGreen}}
\newcommand\DLBa{\color{RoyalBlue}}
\newcommand\DLBb{\color{Cerulean}}
\newcommand\ds{\displaystyle}
\newcommand\ddx{\frac{d}{dx}}
\newcommand\os{\overset}
\newcommand\us{\underset}
\newcommand\ob{\overbrace}
\newcommand\obt{\overbracket}
\newcommand\ub{\underbrace}
\newcommand\ubt{\underbracket}
\newcommand\ul{\underline}
\newcommand\laplacesym{\mathscr{L}}
\newcommand\lap[1]{\laplacesym\left\{#1\right\}}
\newcommand\ilap[1]{\laplacesym^{-1}\left\{#1\right\}}
\newcommand\tikznode[3][]
{\tikz[remember picture,baseline=(#2.base)]
\node[minimum size=0pt,inner sep=0pt,#1](#2){#3};
}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\newcommand{\sfrac}[2]{{#1}/{#2}}
\)
Section 9.1 The LHCC Equation
Before learning how to solve Linear Homogeneous Constant Coefficient (LHCC) differential equations, it’s important to identify these equations and understand their structure. In this section, we’ll define what it means for a differential equation to be homogeneous, review how constant coefficients affect the form of the equation, and formally define what qualifies as an LHCC equation.
Subsection Linear Differential Equations & Forcing Functions
Recall that a differential equation is called
linear if it only contains linear terms, meaning the dependent variable and its derivatives appear to the first power without multiplication or composed with one another.
An equation is
nonlinear if it includes terms like
\(y^2\text{,}\) \((y')^3\text{,}\) \(\sin(y)\text{,}\) or
\(y \cdot y''\text{,}\) where the dependent variable is raised to a power or multiplied by one of its derivatives.
A general \(n\) -th order linear differential equation looks like this:
\begin{equation}
\us{ \large\text{of}\ y^{(n)} }{ \us{ \large\text{coefficient} }{
\us{\uparrow}{ {\color{BurntOrange} \ul{a_n(x)} }}\ y^{(n)}
}} + \cdots +
\us{ \large\text{of}\ y'' }{ \us{ \large\text{coefficient} }{
\us{\uparrow}{ {\color{BurntOrange} \ul{a_2(x)} }}\ y''
}} +
\us{ \large\text{of}\ y' }{ \us{ \large\text{coefficient} }{
\us{\uparrow}{ {\color{BurntOrange} \ul{a_1(x)} }}\ y'
}} +
\us{ \large\text{of}\ y }{ \us{ \large\text{coefficient} }{
\us{\uparrow}{ {\color{BurntOrange} \ul{a_0(x)} }}\ y
}} =
\us{ \large\text{function} }{ \us{ \large\text{forcing} }{
\us{\uparrow}{ {\color{BurntOrange} \ul{f(x)} }}
}}\tag{9.1}
\end{equation}
The coefficients
\(a_n(x), \dots, a_0(x)\) and the right-hand side
\(f(x)\) depend only on the independent variable, which we’re calling
\(x\text{.}\) While we previously referred to
\(f(x)\) as a free term, in this context, it is known as the
forcing function .
Forcing functions are not always easily identifiable on the right-hand side of a differential equation. Sometimes,
\(y\) terms and their derivatives may be mixed in, or parts of the forcing function may appear on the left.
For example, in the equation
\begin{equation*}
y'' + 2y + 5x = -5 + 3y'\text{,}
\end{equation*}
the forcing function is not immediately clear. Rearranging it into standard form:
\begin{equation*}
y'' - 3y' + 2y = 5x - 5
\end{equation*}
reveals that the forcing function is \(5x - 5\text{.}\)
Checkpoint 121 . 📖❓ Identifying the Forcing Function.
What is the forcing function in the differential equation?
\begin{equation*}
x^2y'' - 5x - 4xy' = 10
\end{equation*}
Subsection Constant Coefficients
A linear differential equation has
constant coefficients if all the coefficients
\(a_n, a_{n-1}, \dots, a_0\) are constants rather than functions of the independent variable
\(x\text{.}\)
So, a linear differential equation with constant coefficients looks like:
\begin{equation}
a_{n}\ y^{(n)} + a_{n-1}\ y^{(n-1)} + \cdots + a_2\ y'' + a_1\ y' + a_0\ y = f(x)\text{.}\tag{9.2}
\end{equation}
Here are a few examples of linear equations with constant coefficients:
\begin{equation*}
2y'' - 3y' + 5y = 0, \qquad y''' + 4y'' - 2y' + 7y = e^x,
\end{equation*}
And here are some that do not have constant coefficients:
\begin{equation*}
\boxed{3^x}\ y'' + \boxed{2x}\ y' + y = 0, \qquad y'' + y' + \boxed{\ln x}\ y = \sin x.
\end{equation*}
Checkpoint 122 . 📖❓ Identifying Constant Coefficient Equations.
Select all the linear differential equations that have constant coefficients.
\(\quad y'' - 4y' + 7y = 0 \)
Correct! All coefficients are constant.
\(\quad x^2 y'' + x y' - y = 0 \)
Incorrect. The coefficient of \(y'' \) is \(x^2 \text{,}\) which is not constant.
\(\quad y''' - 2y' + 3y = 0 \)
Correct! All coefficients are constant.
\(\quad e^x y'' + y' + y = 0 \)
Incorrect. The coefficient \(e^x \) is not a constant.
Subsection Homogeneous
A linear differential equation is
homogeneous if the right-hand side is zero—that is, there is no forcing term. Otherwise, it is
nonhomogeneous .
For example, the following equations are homogeneous:
\begin{equation*}
x y'' - \cos x y' + 6y = 0, \qquad y''' + 2y'' = y'
\end{equation*}
These are nonhomogeneous because of the (boxed) right-hand side:
\begin{equation*}
y'' + 4xy = \boxed{x}, \qquad y' + \boxed{e^x} = 0
\end{equation*}
Checkpoint 123 . 📖❓ Identifying Homogeneous Equations.
Which of the following equations is homogeneous?
\(\ds\quad y'' + 4y' + 3y = 0 \) Correct! The right-hand side is zero, so the equation is homogeneous.
\(\ds\quad y'' + 4y^2 = 0 \) Incorrect. This equation appears homogeneous, but homogeneity applies only to linear equations.
\(\ds\quad y'' + 4y' + 3y = 7 \) Incorrect. The equation has a non-zero free term, making it non-homogeneous.
\(\ds\quad y' + 2y = x^2 \) Incorrect. The right-hand side has a non-zero function of \(x\text{,}\) so this is non-homogeneous.
Subsection The LHCC Equation
Putting it all together, we arrive at the definition of an LHCC equation.
📙 Definition 124 .
A linear homogeneous differential equation with constant coefficients (LHCC) has the form:
\begin{equation}
a_n\ y^{(n)} + \cdots + a_2\ y'' + a_1\ y' + a_0\ y = 0\tag{9.3}
\end{equation}
where all coefficients
\(a_n, \dots, a_0\) are constants.
The following equations are examples of LHCC equations:
\begin{equation*}
2y'' - 3y' + 5y = 0, \qquad \frac{d^2s}{dt^2} + \frac{ds}{dt} - 4s = 0 \qquad u'' = u.
\end{equation*}
Checkpoint 125 . 🤔💭 Reading Questions.
(a) 🤔💭 Matching Equations with Properties.
You have attempted
of
activities on this page.
