\chapter{Linear ODEs}

Linear ODEs are those of the following form.

\[\sum^{n}_{k=0} a_k(x) y^{(k)} = b(x)\]

Specifically, homogeneous linear ODEs will be the center of our concentration; the central idea is that solution functions to homogeneous linear ODEs form a linear space! The application of linear algebra is the motivation behind many of the ideas presented in this chapter. Even for inhomogeneous linear ODEs, a general solution still requires the evaluation

\begin{proposition}
The general solution to an inhomogeneous linear ODE is the homogeneous variant's general solution plus some function called the \emph{particular solution}.
\end{proposition}


From a result called the Picard-Lindeloef theorem, we are assured of the following proposition.

\begin{proposition}
Consider the following linear ODE where the $a_k : I_k \to \mathbb{R}$ and $b : I_b \to \mathbb{R}$ are continuous on their domain
\[\sum^{n}_{k=0} a_k(x) y^{(k)} = b(x)\]
This ODE has a general solution on $I_b cup [ \bigcup_{k=0}^{n} I_k] $
The general solution of a linear ODE of degree $n$ is the linear combination of $n$ distinct functions.
\end{proposition}

\begin{corollary}
Consider the following linear ODE where the $a_k : I_k \to \mathbb{R}$ and $b : I_b \to \mathbb{R}$ are continuous on their domain
\[\sum^{n}_{k=0} a_k(x) y^{(k)} = b(x)\]
Let $F = \{f \in C^{n}(I) : \forall x \in I [ \sum^{n}_{k=1} q_k (x) f^{(k)}(x) = 0 ] \}$, then $(F,\mathbb{R},+,\cdot)$ forms a linear space over $\mathbb{R}$ (where addition and scalar multiplication are usual addition and scaling of functions).
\end{corollary}

The fact that solutions to inhomogeneous linear ODEs form a linear space is central, and questions about the properties of this linear space will in turn provide information of the solutions to the ODE.

Due to the solution space being a linear space, we are led to believe that its general solution will be the linear combination of linearly independent solutions.
This begs some questions; what is the dimension of this linear space? How can we find linearly independent solutions?



\section{Wronskian}



\begin{definition}[Wronskian]
Given a vector of real or complex functions $\mathbf{y}$, their \emph{Wronskian} is the function \(W\) which is the determinant of the matrix where each column is a vector with a solution to a linear DE and its \(n-1\) derivatives.
\[ W(\mathbf{y})(x) = \begin{vmatrix} y_1 & y_2 & \cdots & y_n \\ y'_1 & y'_2 & \cdots & y'_n \\ \vdots & \vdots & \ddots & \vdots \\ y^{(n-1)}_{1} & y^{(n-1)}_{2} & \cdots & y^{(n-1)}_{n} \end{vmatrix} \]

Where  $\mathbf{y} = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{bmatrix}$ is a vector of solutions to a linear DE
\end{definition}


\[\mathbf{y} \text{ are linearly dependent on } I \implies W(I) =\{0\} \]
Where $ \mathbf{y} = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{bmatrix}$ is a vector of solutions to a linear DE


\[\mathbf{y} \text{ are analytic on } I \implies [ \mathbf{y} \text{ are linearly dependent on } I \iff W(I) =\{0\} ] \]
\[\mathbf{y} \text{ satisfy } y''+q_1 (x)y'+q_2(x)=0 \text{ on } I \implies [ \mathbf{y} \text{ are linearly dependent on } I \iff W(I) =\{0\} ] \]

\subsection{Abel's formula}
\begin{theorem}[Abel's formula]
Let $y_1,y_2$ be functions satisfying $y''+ a_1 (x)y' + a_2 (x) y=0$, then $W(y_1 , y_2) = K_{12}e^{-\int q_1 (x) dx}$
%\[ y_1,y_2 \text{ satisfy } y'' + q_1 (x) y' + q_2 (x) y = 0  \implies W(y_1 , y_2) = K_{12}e^{-\int q_1 (x) dx}\]
\end{theorem}

Now that we have a way of calculating the Wronskian in a way that is independent on knowing both functions, we can do something interesting; we can derive the second function purely from the first!

\begin{corollary}
Let $y_1$ be a nonvanishing function satisfying $y''+ a_1 (x)y' + a_2 (x) y=0$, then the following function $y_2$  is linearly independent to $y_1$ and also satisfies the ODE.
	\[ y_2 = y_1 \int \frac{e^{-\int q_1 (x) dx}}{y_1 (x)^2} dx\]
%\[ y_1 \text{ satisfies } y'' + q_1 (x) y' + q_2 (x) y = 0  \land \neg ( \forall x [y_1(x)=0] )  \implies y_2 = y_1 \int \frac{e^{-\int q_1 (x) dx}}{y_1 (x)^2} dx \text{ satisfies } y'' + q_1 (x) y' + q_2 (x) y = 0 \land y_1 , y_2 \text{ are linearly independent} \]
\end{corollary}

As wonderful as it is that we can abuse the linear space formed by the linear ODE to generate novel solutions using known solutions, this still requires that we know some solutions for the ODE.

To get an initial foothold on the problem, the Annihilator method or Cauchy-Euler method can be used to obtain a first set of solutions before expanding our solutions using Abel's formula. Unfortunately such methods are not always possible to apply due to assumptions not being met, but there exists some more potent methods for evaluating solutions to linear ODEs.




\section{Variation of Parameters (VOP)}

A general technique to solve inhomogeneous linear ODEs is VOP.
VOP is essentially just assuming the ansatz $y(x) = u(x)y_1(x) + v(x) y_2 (x)$ and $u' y_1 + v' y_2 = 0$



\section{Series methods}


One powerful idea is to assume that the solution to the linear ODE is an analytic function, meaning that we assume the ansatz $y(x) = \sum^{\infty}_{n=0} a_n x^n$. It's quite a simple idea, yet incredibly effetive since the class of analytic functions is quite rich indeed. Often the solutions to an ODE are nonelementary, so this method comes in clutch where others fail.







For differential equations such that $y^{(n)} + \sum q_k (x) y^{(k)} = 0 \land q_k \text{ are analytic on }I \land 0 \in I$, solutions are always analytic and hence has a Taylor series representation and hence a power series representation. This is due to considering the DE on a complex neighborhood and by noting that holomorphic functions are analytic, see Complex Analysis.
\begin{itemize}
	\item Assume the ansatz $y(x) = \sum^{\infty}_{n=0} a_n x^n$
	\item Substitute the ansatz into the DE and manipulate into one single sum
			$ [ \sum^{\infty}_{n=0} a_n x^n ]^{(k)} = \sum^{\infty}_{n=k} a_n \frac{n!}{(n-k)!} x^{n-k}$
			<li>Since the RHS equals zero, equate the coefficient of the new sum to 0<
			<li>Solve recursive formula for $a_n$
			</ul>

\begin{definition}[Ordinary point]
	\[ x_0 \text{ is an ordinary point of } \sum q_n (x) y^{n} (x) = 0 \iff \forall k \in \{0,1,..,n\}a_k \text{ is analytic at }x_0) \]
\end{definition}

Translating from a recurrence relation into a more recognizable notation can be a bit laborious, however a few elementary identities could help out.

\[ \prod^{n}_{k=1} 2k = 2^n n! \]
\[ \prod^{n}_{k=1} (2k-1) = \frac{(2n-1)!}{2^(n-1) (n-1)!}\]

\begin{definition}[Singular point]
\[x_0 \text{ is a singular point of } \sum q_n (x) y^{n} (x) = 0 \iff \neg [ x_0 \text{ is a singular point of } \sum q_n (x) y^{n} (x) = 0]\]
\end{definition}
All coefficient function analytic at a point means all DEs solutions taylor series equal function itself



Series method
Method of Frobenius



Additionally, the Laplace transform can be used to transform ODEs into algebraic equations! Here is the outline of the method.
\begin{itemize}
	\item Compute the Laplace transform for both sides of the equation and equate them
	\item Solve algebraically for $y_1$
	\item Compute $\mathcal{L}^{-1}\{y_1\}(x)$
\end{itemize}



\begin{definition}[Confluent hypergeometric functions]
Linearly independent solutions to the ODE \(zw''+(b-z)w'-aw=0\) (Kummer's equation)
First kind
\[ M(a,b,z)= \sum^{\infty}_{n=0} \frac{a^{(n)} z^n }{ b^{(n)} n! } \]
\begin{itemize}
	\item $x^{(n)}=\prod^{n-1}_{k=0}(x+k)$ is the rising factorial
	\item $ (x)_{n}=\prod^{n-1}_{k=0}(x-k)$ is the falling factorial
\end{itemize}
Second kind
\[ U(a,b,z) \]
\end{definition}







\chapter{PDEs}


\begin{definition}[Partial differential equation (PDE)]
A \emph{partial differential equation (PDE)} is an equation of the following form, defined by some $f : S \subseteq \mathbb{R}^n \times \mathbb{R}^m$ and $D^k \mathbf{y}$ representing the matrix of possible partial derivatives of order $k$.
\[ f(\mathbf{x},y(\mathbf{x}), D y(\mathbf{x}) , \hdots , D^m y(\mathbf{x})) = 0\]
Solutions for $y : U \to \mathbb{R}$ are (usually, depending on whether all variables can be differentiated $m$ times) in $C^{m}(U)$, for some set $U$ open in (Euclidean topology) $\mathbb{R}^m$.
\end{definition}

The introduction of a multivariable function already seriously complicates the formal definition from an ODE, and complication is generally the name of the game with PDEs.

This chapter aims to look at solution methods on the more elementary side that one can apply in the study of PDEs, generally mapping a PDE to multiple ODEs.


\section{Separation of variables}

The most elementary solution method I know of is \emph{separation of variables}, which basically makes a general yet bold ansatz that is an attempt to split our PDE into ODEs with respect to one variable each, then prays to the math gods that things work out.

This solution method has a chance at being succesful when the RHS (or any 1 of the sides) of the PDE reduces to a constant after we make our ansatz because this allows us to split the PDE into multiple ODEs. Here is the general method.

\begin{itemize}
	\item Assume the ansatz $f(x,y) = X(x)Y(y)$ and substitute into equation
	\item Separate variables to both sides and equate both sides to a constant $\pm \lambda$
	\item Solve each problem using SL theory or otherwise and then substitute the solutions into the original ansatz; the general solution is a linear combination of these eigenfunctions
	\item Equate the general solution to the final boundary condition and find the projection.
\end{itemize}



\section{Fourier transform method}

Using the ansatz (guess) that a solution can be represented as the Fourier transform of some function $u$ is quite effective.

Fourier studied a PDE now hailed as the heat equation, and he tried to find new solutions to it based on the trivial , which were sinudoids. His idea was basically to consider convolutions of some unknown function with these elementary solutions to find further solutions.

Fourier conjectured that (sufficiently nice) periodic functions could always be represented as something called a Fourier series, and that non-periodic function (again, sufficiently nice ones, whatever that means) could still be represented by a 'Fourier integral'. Though such statements could not be verified until much later, Fourier's steadfast beliefs ultimately paid off, producing legitimate solutions to the heat equation.

Fourier analysis warrant much attention due to its usefulness in PDEs as well as signal processing, and though we do not discuss the theory here, you can read about it in Fourier Analysis. To those already familiar with the theory, I offer you a neat way to solve PDEs with Fourier analysis.



\section{Qualitative theorems}
\subsection{Maximum principle}
\subsection{Harnack's theorem}