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\begin{document}


\section{Equations with Constant Coefficients}

A linear differential equation of order $n$ with constant coefficients has
the form 
\[
a_{0}\frac{d^{n}y}{dx^{n}}+a_{1}\frac{d^{n-1}y}{dx^{n-1}}+\cdots +a_{n-1}%
\frac{dy}{dx}+a_{n}y=b(x) 
\]
where $a_{0},a_{1},\ldots ,a_{n}$ are real constants such that $a_{0}\neq 0$
and $b$ is a real-valued function that is continuous on a real interval $%
c\leq x\leq d$. The corresponding homogeneous equation is given by 
\[
a_{0}\frac{d^{n}y}{dx^{n}}+a_{1}\frac{d^{n-1}y}{dx^{n-1}}+\cdots +a_{n-1}%
\frac{dy}{dx}+a_{n}y=0\text{.} 
\]

\subsection{Homogeneous Equations with Constant Coefficients\label%
{Homogenous Equations}}

\begin{definition}
The \textbf{characteristic polynomial\label{Characteristic Polynomial}} is
defined by 
\[
\Delta (\lambda )=a_{0}\lambda ^{n}+a_{1}\lambda ^{n-1}+\cdots
+a_{n-1}\lambda +a_{n}\text{.}
\]
\end{definition}

The characteristic polynomial can be used to generate $n$ explicit functions
which form a fundamental set of solutions. The general solution is then a
linear combination of these $n$ solutions.

Using the Fundamental Theorem of Algebra, the polynomial $\Delta $ can be
completely factored as 
\[
\Delta (\lambda )=a_{0}(\lambda -\lambda _{1})^{m_{1}}(\lambda -\lambda
_{2})^{m_{2}}\cdots (\lambda -\lambda _{s})^{m_{s}}\text{.} 
\]
The complex numbers $\lambda _{1},\lambda _{2},\ldots \lambda _{s}$ are the
distinct roots of the \textbf{characteristic} \textbf{equation} \label%
{Characteristic Equation}$\Delta (\lambda )=0$ and the positive integer $%
m_{k}$ is called the \textbf{multiplicity} of the root $\lambda _{k}$ for $%
k=1,\ldots ,s$. Observe that $n=m_{1}+\cdots m_{s}$. Since the coefficients
of the characteristic polynomial are real, the complex roots of the
characteristic equation occur in conjugate pairs; that is, if the number $%
a+bi$, where $i$ is the imaginary unit, is a root of the characteristic
equation, then so is its complex conjugate $a-bi$.

\begin{theorem}
Let $\lambda _{1},\lambda _{2},\ldots \lambda _{s}$ be the distinct roots of
the characteristic equation 
\[
a_{0}\lambda ^{n}+a_{1}\lambda ^{n-1}+\cdots +a_{n-1}\lambda +a_{n}=0 
\]
and suppose $\lambda _{k}$ has multiplicity $m_{k}$ for $k=1,\ldots ,s$.
Then the fundamental set of solutions is given by the $n$ functions 
\[
\begin{tabular}{lrll}
& $x^{j}e^{\lambda _{k}x}$ & $(j=0,\ldots ,m_{k}-1)$ & $\text{if }\lambda
_{k}\text{ is real}$ \\ 
$\text{and }$ & \multicolumn{1}{l}{$\text{\ }x^{j}e^{c_{k}x}\cos (d_{k}x)%
\text{ , }x^{j}e^{c_{k}x}\sin (d_{k}x)$} & $(j=0,\ldots ,m_{k}-1)$ & $\text{%
if }\lambda _{k}=c_{k}+id_{k},d_{k}>0\text{.}$%
\end{tabular}
\]
\end{theorem}

\subsection{Nonhomogeneous Equations with Constant Coefficients\label%
{Nonhomogeneous Equations}}

The \hyperref{Variation of Parameters \label{Variation of Parameters}}{}{}{%
ODE02_01.tex#Variation of Parameters}formula can always be used to find a
particular solution once a fundamental set of solutions has been determined.
This formula can be interpreted as the result of solving a certain system of
linear algebraic equations. Using this approach, the following theorem is a
restatement of the variation of parameters method for finding a particular
solution.

\begin{theorem}
Let $f_{1},f_{2},\ldots ,f_{n}$ be a fundamental set of solutions and
consider the system of linear algebraic equations 
\[
\left( 
\begin{array}{c}
0 \\ 
0 \\ 
\vdots \\ 
\frac{b(x)}{a_{0}}
\end{array}
\right) =\left( 
\begin{array}{cccc}
f_{1}(x) & f_{2}(x) & \cdots & f_{n}(x) \\ 
f_{1}^{(1)}(x) & f_{2}^{(1)}(x) & \cdots & f_{n}^{(1)}(x) \\ 
\vdots & \vdots & \vdots & \vdots \\ 
f_{1}^{(n-1)}(x) & f_{2}^{(n-1)}(x) & \cdots & f_{n}^{(n-1)}(x)
\end{array}
\right) \left( 
\begin{array}{c}
u_{1}(x) \\ 
u_{2}(x) \\ 
\vdots \\ 
u_{n}(x)
\end{array}
\right) \text{.} 
\]
Then this system of equations uniquely determines $n$ functions $%
u_{1},u_{2},\ldots u_{n}$. Futhermore, if $n$ functions $v_{1},v_{2},\ldots
v_{n}$ are antiderivatives of $u_{1},u_{2},\ldots u_{n}$, respectively, then
the function 
\[
v=v_{1}f_{1}+v_{2}f_{2}+\cdots v_{n}f_{n} 
\]
is a particular solution.
\end{theorem}

Another method for finding a particular solution, which only applies to
linear differential equations with constant coefficients, is the \textbf{%
method of} \textbf{undetermined coefficients\label%
{Method of Undetermined Coefficients}}. This method works only if the
function $b$ in has a particular form.

\begin{definition}
A \textbf{UC function} is a function defined as the finite product of one or
more of the following:

\begin{enumerate}
\begin{enumerate}
\begin{enumerate}
\item  $x^{n}$, where $n$ is a non-negative integer.

\item  $e^{ax}$, where $a$ is a non-zero constant.

\item  $\sin bx$, where $b$ is a non-zero constant.

\item  $\cos bx$, where $b$ is a non-zero constant.
\end{enumerate}
\end{enumerate}
\end{enumerate}
\end{definition}

\begin{theorem}
Derivatives of UC functions are finite linear combinations of UC functions.
\end{theorem}

Suppose a linear differential equation with constant coefficients has a
right-hand side has the form $cu\left( x\right) $, where $c$ is a constant
and $u\left( x\right) $ is a UC function. The following procedure will
determine a particular solution.

\begin{enumerate}
\item  Find the set consisting of $u$ and all linearly independent UC
functions of which the successive derivatives of $u$ are linear
combinations. Call this set $S$.

\item  If $S$ contains a solution of the corresponding homogeneous equation,
then multiply each member of $S$ by the smallest positive power of $x$ so
that the revised set no longer contains a solution. Replace $S$ with this
revised set.

\item  Form a linear combination of all of the members of $S$ with unknown
coefficients (\textit{undetermined} \textit{coefficients}).

\item  Substitute this linear combination for $y$ in the left-side of the
nonhomogeneous equation.

\item  After performing the differentiations, collect like terms and equate
coefficients with the right-hand side. Solve the resulting system of linear
algebraic equations to determine the coefficients of the linear combination.
\end{enumerate}

If the right-hand side $b$ is any linear combination of UC functions, then
the above procedure can be applied to each term of the linear combination
and then the \hyperref{Superposition Principle}{}{}{%
ODE02_01.TEX#Superposition Principle} can be applied to find a particular
solution.

\noindent \hrulefill

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\noindent \hrulefill

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