\chapter{Differential equations}

By introducing the derivative, we also create the notion of \emph{differential equations}.

They are extemely powerful at capturing physical phenomena, though they are not always so easy to solve.

\section{Types of differential equations}
linear
order
degree
IVP

general solution
particular solution
\section{Infinitesimal generator}

Before solving any ODEs, we'll first look at a really powerful qualitative device for first order ODEs; the \emph{infinitesimal generator}. It is essentially a field of vectors in space that demonstrates how a solution to a first order ODE evolves for small progressions in the domain. In other words, it shows how a solution is "generated" when taking "infinitesimal" steps.


\begin{definition}[Infinitesimal generator of first order ODE]
$y' = f(x,y)$
$x' = 1$
$\mathbf{F}(x,y) = \begin{pmatrix} f(x,y) \\ x \end{pmatrix}$
\end{definition}

Here we are using infinitesimal generators as a way to view solutions of a differential equation from a geometric perspective, though in advanced mathematics (specifically Lie theory and differential geometry), infinitesimal generators become extremely important objects.



\section{Solution methods}
\subsection{Separating equation}


We will discuss a general method to solve first order seperable ODEs.
\[y'=f(x)g(y)\]
\[\frac{y'}{g(y)}=f(x)\]
\[\int \frac{1}{g(y)} dy=\int f(x) dx\]

Now one calculates the integrals and solves algebraically for $y$ to obtain a solution.


\subsection{Integrating factor}


\subsection{Second order linear homogeneous  ODE with constant coefficients}


\[ay'' +by'+cy =0 \]