\newsectionNoPB
\subsection{Linear Differential Equations}
An ODE is considered linear if and only if the $y$s are only scaled and not part of powers.\\
\compactdef{Linear differential equation of order $k$} (order = highest derivative)
$y^{(k)} + a_{k - 1}y^{(k - 1)} + \ldots + a_1 y' + a_0 y = b$, with $a_i$ and $b$ functions in $x$.
If $b(x) = 0 \smallhspace \forall x$, \bi{homogeneous}, else \bi{inhomogeneous}\\
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\shorttheorem For open $I \subseteq \R$ and $k \geq 1$, for lin. ODE over $I$ with continuous $a_i$ we have:
\rmvspace
\begin{enumerate}[noitemsep]
    \item Set $\mathcal{S}$ of $k \times$ diff. sol. $f: I \rightarrow \C (\R)$ of the eq. is a complex (real) subspace of complex (real)-valued func. over $I$
    \item $\dim(\mathcal{S}) = k \smallhspace\forall x_0 \in I$ and any $(y_0, \ldots, y_{k - 1}) \in \C^k$, exists unique
          $f \in \mathcal{S}$ s.t. $f(x_0) = y_0, f'(x_0) = y_1, \ldots, f^{(k - 1)}(x_0) = y_{k - 1}$.
          If $a_i$ real-valued, same applies, but $\C$ replaced by $\R$.
    \item Let $b$ continuous on $I$. Exists solution $f_0$ to inhom. lin. ODE and $\mathcal{S}_b$ is set of funct. $f + f_0$ where $f \in \mathcal{S}$
\end{enumerate}
The solution space $\mathcal{S}$ is spanned by $k$ functions, which thus form a basis of $\mathcal{S}$. If inhomogeneous, $\mathcal{S}$ not vector space.

\shade{gray}{Finding solutions (in general)}
\rmvspace
\begin{enumerate}[label=\bi{(\arabic*)}, noitemsep]
    \item Find basis $\{ f_1, \ldots, f_k \}$ for $\mathcal{S}_0$ for homogeneous equation (set $b(x) = 0$) (i.e. find homogeneous part, solve it)
    \item If inhomogeneous, find $f_p$ that solves the equation. The set of solutions is then $\mathcal{S}_b = \{ f_h + f_p \divides f_h \in \mathcal{S}_0 \}$.
    \item If there are initial conditions, find equations $\in \mathcal{S}_b$ which fulfill conditions using SLE (as always)
\end{enumerate}
