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Section The LHCC Equation
Before learning how to solve Linear Homogeneous Constant Coefficient (LHCC) differential equations, itβs important to identify these equations and understand the nature of their structure. In this section, weβll 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
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{31}
\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 weβre 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, the equation
\begin{equation*}
y'' + 2y + 5x = -5 + 3y'
\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 128 . πβ 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{32}
\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\text{,}
\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\text{.}
\end{equation*}
Checkpoint 129 . πβ 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 zeroβthat 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 130 . πβ 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 looks homogeneous, but homogeneity only applies 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.
π Definition 131 .
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{33}
\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\text{.}
\end{equation*}
Subsection π€ Wrap-Up
ποΈ \(\textbf{Key Takeaways...}\)
The forcing function is the collection of all terms that are not multiplied by the dependent variable, typically written on the right-side.
An equation has constant coefficients when all the coefficients multiplying \(y\) and its derivatives are fixed constants (not functions of \(x\) ).
A differential equation is homogeneous if the forcing function is zero.
An \(\DLO \text{LHCC}\) equation is a \(\DLO \text{L}\) inear \(\DLO \text{H}\) omogeneous \(\DLO \text{C}\) onstant \(\DLO \text{C}\) oefficient equation.
Check Your Understanding.
Checkpoint 132 . π€π Separation of Variables Reading Questions.
(a) π€π Matching Equations with Properties.
You have attempted
of
activities on this page.
