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\begin{document}


\section{Basic Theory of Linear Differential Equations}

\begin{definition}
\vspace{1pt}A \textbf{linear differential equation of order\label%
{Linear Differential Equation} }$n$ is an equation of the form 
\[
a_{0}(x)\frac{d^{n}y}{dx^{n}}+a_{1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdots
+a_{n-1}(x)\frac{dy}{dx}+a_{n}(x)y=b(x)\text{.}
\]
\end{definition}

Throughout this section it is assumed that the functions $a_{0},a_{1},\ldots
,a_{n}$ and $b$ are real-valued and continuous on a real interval $I:c\leq
x\leq d$ and $a_{0}(x)\neq 0$ for all $x\in I$.

The functions $a_{0},a_{1},\ldots ,a_{n}$ are called the \textbf{coefficients%
}. The right hand member $b(x)$ is called the \textbf{nonhomogeneous term}.
If $b$ is identically zero, then the equation reduces to 
\[
a_{0}(x)\frac{d^{n}y}{dx^{n}}+a_{1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdots
+a_{n-1}(x)\frac{dy}{dx}+a_{n}(x)y=0 
\]
and is then called \textbf{homogeneous}. The term \textit{homogenous} has a
different meaning for first-order differential equations(see the topic 
\hyperref{Separable and Homogenous Equations}{}{}{ODE01_02.tex}). If the
leading coefficient $a_{0}$ is identically $1$, then the equation is said to
be \textbf{normalized}. Since $a_{0}(x)\neq 0$ for all $x\in I$, the
equation can always be normalized by dividing both sides by $a_{0}$.

A \textbf{solution} to the differential equation is a real-valued function $%
f $ defined on the interval $I$ which possesses $n$ derivatives and
satisfies the equation for all $x\in I$. The associated initial-value
problem is stated as follows:

\begin{description}
\item[Initial-value Problem]  \label{Initial-value Problem}Let $x_{0}\in I$
and let $d_{0},d_{1},\ldots ,d_{n-1}$ be $n$ arbitrary constants. Find a
solution $f$ such that $f^{(k)}(x_{0})=d_{k}$ for $k=0\ldots n-1$.
\end{description}

\subsection{Existence and Uniqueness of Solutions\label%
{Existence and Uniqueness}}

The following result on the solutions of the initial-value problem plays a
pivotal role in proving many of the remaining theorems of this section.

\begin{theorem}
Let $x_{0}\in I$ and let $d_{0},d_{1},\ldots ,d_{n-1}$ be $n$ real
constants. Then there exists a unique solution $f$ of the nonhomogeneous
equation which satisfies $f^{(k)}(x_{0})=d_{k}$ for $k=0\ldots n-1$.
\end{theorem}

\subsection{Homogeneous Equations\label{Homogeneous Equations}}

Since the zero function on $I$ is clearly a solution of the homogeneous
equation, the following theorem is a simple consequence of the uniqueness
theorem.

\begin{theorem}
Let $x_{0}\in I$ and let $f$ be a solution of homogeneous equation such that 
$f^{(k)}(x_{0})=0$ for $k=0\ldots n-1$. Then $f(x)=0$ for all $x\in I$.
\end{theorem}

\begin{definition}
A \textbf{linear combination} of $m$ functions $f_{1},f_{2},\ldots f_{m}$ is
an expression of the form 
\[
c_{1}f_{1}+c_{2}f_{2}+\cdots +c_{m}f_{m}
\]
where $c_{1},c_{1},\ldots ,c_{m}$ are constants.
\end{definition}

\begin{definition}
The functions $f_{1},f_{2},\ldots ,f_{m}$ , all defined on an interval $I$,
are called \textbf{linearly dependent on }$I$ \label%
{Linearly Dependent Functions}if there exist constants $c_{1},c_{2},\ldots
c_{m}$, \textbf{not all zero}, such that 
\[
c_{1}f_{1}(x)+c_{2}f_{2}(x)+\cdots +c_{m}f_{m}(x)=0\text{ for all }x\in I%
\text{.}
\]
\end{definition}

\begin{definition}
The functions $f_{1},f_{2},\ldots ,f_{m}$ , all defined on an interval $I$,
are called \textbf{linearly independent on }$I\label%
{Linearly independent funcitions}$ if they are not linearly dependent there;
that is, the relation 
\[
c_{1}f_{1}(x)+c_{2}f_{2}(x)+\cdots +c_{m}f_{m}(x)=0\text{ for all }x\in I%
\text{.}
\]
implies that $c_{1}=c_{2}=\cdots =c_{n}=0$.
\end{definition}

\begin{theorem}
Any linear combination of solutions of the homogeneous equation is also a
solution.
\end{theorem}

\begin{theorem}
The $nth$ order homogeneous equation always possesses $n$ linearly
independent solutions $f_{1},f_{2},\ldots ,f_{n}$ on the interval $I$, and
every solution can be expressed as a linear combination of these $n$
linearly independent solutions.
\end{theorem}

Any set of $n$ linearly independent solutions $f_{1},f_{2},\ldots ,f_{n}$ is
called a \textbf{basis} or a \textbf{fundamental set\label%
{Fundamental Set of Solutions} }of solutions. The linear combination $%
c_{1}f_{1}+c_{2}f_{2}+\cdots +c_{n}f_{n}$, where $c_{1},c_{2},\ldots c_{n%
\text{ }}$are arbitrary constants, is called the \textbf{general solution}
(relative to the basis functions $f_{1},f_{2},\ldots ,f_{n}$).

\begin{definition}
Let $f_{1},f_{2},\ldots ,f_{n}$ be $n$ solutions. The \textbf{Wronskian} 
\label{Wronskian}with respect to $f_{1},f_{2},\ldots ,f_{n}$ , denoted by $%
W\left( f_{1},f_{2},\ldots ,f_{n}\right) $, is defined as the determinant of
the following matrix: 
\[
\left( 
\begin{array}{cccc}
f_{1} & f_{2} & \cdots  & f_{n} \\ 
f_{1}^{(1)} & f_{2}^{(1)^{{}}} & \cdots  & f_{n}^{(1)} \\ 
\vdots  & \vdots  & \cdots  & \vdots  \\ 
f_{1}^{(n-1)} & f_{2}^{(n-1)} & \cdots  & f_{n}^{(n-1)}
\end{array}
\right) \text{.}
\]
Observe that $W\left( f_{1},f_{2},\ldots ,f_{n}\right) $ is a function
defined on the interval $I$.
\end{definition}

\begin{theorem}
Let $x_{0}\in I$ and let $f_{1},f_{2},\ldots ,f_{n}$ be $n$ solutions. Then 
\[
W\left( f_{1},f_{2},\ldots ,f_{n}\right) (x)=W\left( f_{1},f_{2},\ldots
,f_{n}\right) (x_{0})\exp \left[ \int_{x_{0}^{{}}}^{x}-\frac{a_{1}(t)}{%
a_{0}(t)}dt\right] \text{ for all }x\in I\text{.}
\]
Therefore, the Wronskian is either identically zero on $I$ or else is never
zero on $I$.
\end{theorem}

The next theorem gives a simple criterion for determining whether or not $n$
solutions are linearly independent and hence form a fundamental set of
solutions.

\begin{theorem}
The $n$ solutions $f_{1},f_{2},\ldots ,f_{n}$ are linearly independent on $I$
if and only if $W\left( f_{1},f_{2},\ldots ,f_{n}\right) (x)\neq 0$ for some 
$x\in I$.
\end{theorem}

\begin{theorem}
Suppose $f_{1},f_{2},\ldots ,f_{n}$ are $n$ functions which possess
continuous $n^{th}-$order derivatives on an interval $I$ and suppose $%
W\left( f_{1},f_{2},\ldots ,f_{n}\right) (x)\neq 0$ for all $x\in I$. Then
there exists a unique normalized homogeneous linear differential equation of
order $n$ for which these functions form a fundamental set. This equation is
given by 
\[
\frac{W\left( f_{1},f_{2},\ldots ,f_{n},y\right) (x)}{W\left(
f_{1},f_{2},\ldots ,f_{n}\right) (x)}=0\text{.}
\]
\end{theorem}

\begin{theorem}
\textbf{(Reduction of Order)} \ \label{Reduction of Order}Let $f$ be a
nontrivial solution. Then the transformation $y=fu$ reduces the homogeneous
equation to an $(n-1)st$ order homogeneous linear differential equation in
the dependent variable $w=\frac{du}{dx}$. Futhermore, let $%
w_{1},w_{2},\ldots w_{n-1}$ be $n-1$ linearly independent solutions of the
reduced equation and, for any $x_{0}\in I$, define functions $%
u_{1},u_{2},\ldots u_{n-1}$ by $u_{k}(x)=\int_{x_{0}}^{x}w_{k}(t)dt$ for $%
k=1\ldots n-1$. Then the functions $f,fu_{1},fu_{2},\ldots ,fu_{n-1}$ are
linearly independent on the interval $I$ and thus form a fundamental set of
solutions.
\end{theorem}

\subsection{Nonhomogeneous Equations\label{Nonhomogeneous Equations}}

\begin{theorem}
Let $v$ and $w$ be any two solutions of the nonhomogenous equation. Then
their difference is a solution of the corresponding homogeneous equation.
\end{theorem}

\begin{theorem}
Let $y_{p}$ be any particular solution of the nonhomogenous equation and let 
$f_{1},f_{2},\ldots ,f_{n}$ be a fundamental set of solutions of the
corresponding homogeneous equation. Then every solution of the
nonhomogeneous equation has the form 
\[
c_{1}f_{1}+c_{2}f_{2}+\cdots +c_{n}f_{n}+y_{p}
\]
where $c_{1},c_{2},\ldots c_{n\text{ }}$ are real constants. This expression
is called the \textbf{general solution}. The general solution of the
homogeneous equation is sometimes called the \textbf{complementary } \textbf{%
function\label{Complementary Function}}. Therefore, the general solution of
the nonhomogenous equation is equal to the sum of the complementary function
and a particular solution.
\end{theorem}

The following theorem provides a formula for constructing a particular
solution of the nonhomogeneous equation.

\begin{theorem}
\textbf{(Variation of Parameters)} \ \label{Variation of Parameters}Let $%
f_{1},f_{2},\ldots ,f_{n}$ be a fundamental set of solutions and let $%
x_{0}\in I$. Then the unique solution $h$ of the equation which satisfies $%
h_{{}}^{(k)}(x_{0})=0$ for $k=0\ldots n-1$ is given by 
\[
h(x)=\sum_{k=1}^{n}f_{k}(x)\int_{x_{0}}^{x}\left[ \frac{W_{k}\left(
f_{1},f_{2},\ldots ,f_{n}\right) (t)}{W\left( f_{1},f_{2},\ldots
,f_{n}\right) (t)}\right] \frac{b(t)}{a_{0}(t)}dt
\]
where $W_{k}\left( f_{1},f_{2},\ldots ,f_{n}\right) (t)$ is the determinant
obtained from $W\left( f_{1},f_{2},\ldots ,f_{n}\right) (t)$ by replacing
the $kth$ column by $(0,0,\ldots ,1)$.
\end{theorem}

\begin{theorem}
Let $x_{0}\in I$ and let $d_{0},d_{1},\ldots ,d_{n-1}$ be $n$ arbitrary real
constants. Then the unique solution $f$ of the nonhomogeneous equation
satisfying the initial conditions $f^{(k)}(x_{0})=d_{k}$ for $k=0,\ldots ,n-1
$ is given by 
\[
f(x)=g(x)+h(x)
\]
where $h$ is the function defined above and $g$ is the unique solution of
the corresponding homogeneous equation satisfying the initial conditions $%
g^{(k)}(x_{0})=d_{k}$ for $k=0\ldots n-1$. The function $g$ is given by 
\[
g(x)=c_{1}f_{1}(x)+c_{2}f_{2}(x)+\cdots +c_{n}f_{n}(x)
\]
where the constants $c_{1},c_{2},\ldots c_{n\text{ }}$ are determined by
solving the following system of linear algebraic equations: 
\[
\left( 
\begin{array}{c}
d_{0} \\ 
d_{1} \\ 
\vdots  \\ 
d_{n-1}
\end{array}
\right) =\left( 
\begin{array}{cccc}
f_{1}(x_{0}) & f_{2}(x_{0}) & \cdots  & f_{n}(x_{0}) \\ 
f_{1}^{(1)}(x_{0}) & f_{2}^{(1)}(x_{0}) & \cdots  & f_{n}^{(1)}(x_{0}) \\ 
\vdots  & \vdots  & \vdots  & \vdots  \\ 
f_{1}^{(n-1)}(x_{0}) & f_{2}^{(n-1)}(x_{0}) & \cdots  & f_{n}^{(n-1)}(x_{0})
\end{array}
\right) \left( 
\begin{array}{c}
c_{1} \\ 
c_{2} \\ 
\vdots  \\ 
c_{n}
\end{array}
\right) \text{.}
\]
\end{theorem}

\begin{theorem}
\textbf{(Superposition Principle)} \ \label{Superposition Principle}Let $f$
be a solution of the equation 
\[
\ a_{0}(x)\frac{d^{n}y}{dx^{n}}+a_{1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdots
+a_{n-1}(x)\frac{dy}{dx}+a_{n}(x)y=b_{1}(x)
\]
and let $g$ be a solution of the equation 
\[
a_{0}(x)\frac{d^{n}y}{dx^{n}}+a_{1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdots
+a_{n-1}(x)\frac{dy}{dx}+a_{n}(x)y=b_{2}(x)\text{.}
\]
Then $f+g$ is a solution of the equation 
\[
a_{0}(x)\frac{d^{n}y}{dx^{n}}+a_{1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdots
+a_{n-1}(x)\frac{dy}{dx}+a_{n}(x)y=b_{1}(x)+b_{2}(x)\text{.}
\]
\end{theorem}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Exact solutions}{}{}{../../Help/DM10-1.tex#Differential
equation}}

{\small \hyperref{Solve ODE, exact}{}{}{../../Help/DM10-1.tex#Solve ODE}}

{\small \hyperref{Initial value problems}{}{}{../../Help/DM10-3.tex#Initial
value problems}}

\noindent \hrulefill

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