\newsectionNoPB
\subsection{Linear differential equations with constant coefficients}
The coefficients $a_i$ are constant functions of form $a_i(x) = k$ with $k$ constant, where $b(x)$ can be any function.\\
%
\shade{gray}{Homogeneous Equation}\rmvspace
\begin{enumerate}[noitemsep]
    \item Find \bi{characteristic polynomial} (of form $\lambda^k + a_{k - 1} \lambda^{k - 1} + \ldots + a_1 \lambda + a_0$ for order $k$ lin. ODE with coefficients $a_i \in \R$).
    \item Find the roots of polynomial. The solution space is given by $\{ C_j \cdot x^{v_j - 1} e^{\gamma_i x} \divides v_j \in \N, \gamma_i \in \R \}$
          where $v_j$ is the multiplicity of the root $\gamma_i$ and $C_j$ is a constant.
          For $\gamma_i = \alpha + \beta i \in \C$, we have $C_1 \cdot e^{\alpha x}\cos(\beta x)$, $C_2 \cdot e^{\alpha x}\sin(\beta x)$,
          representing the two complex conjugated solutions.
\end{enumerate}
\rmvspace

The homogeneous equation will then be all the elements of the set summed up.\\
\shade{gray}{Inhomogeneous Equation}\rmvspace
\begin{enumerate}[noitemsep]
    \item \bi{(Case 1)} $b(x) = c x^d e^{\alpha x}$, with special cases $x^d$ and $e^{\alpha x}$:
          $f_p = Q(x) e^{\alpha x}$ with $Q$ a polynomial with $\deg(Q) \leq j + d$,
          where $j$ is multiplicity of root $\alpha$ (if $P(\alpha) \neq 0$, then $j = 0$) of characteristic polynomial
    \item \bi{(Case 2)} $b(x) = c x^d \cos(\alpha x)$, or $b(x) = c x^d \sin(\alpha x)$:
          $f_p = Q_1(x) \cdot \cos(\alpha x) + Q_2(x) \cdot \sin(\alpha x))$,
          where $Q_i(x)$ a polynomial with $\deg(Q_i) \leq d + j$,
          where $j$ is the multiplicity of root $\alpha i$ (if $P(\alpha i) \neq 0$, then $j = 0$) of characteristic polynomial
    \item \bi{(Case 3)} $b(x) = c e^{\alpha x} \cos(\beta x)$, or $b(x) = c e^{\alpha x} \sin(\beta x)$, use the Ansatz
          $Q_1(x) e^{\alpha x} \sin(\beta x) + Q_2(x) e^{\alpha x} \cos(\beta x)$, again with the same polynomial.
          Often, it is sufficent to have a polynomial of degree 0 (i.e. constant)
\end{enumerate}
\rmvspace

For inhomogeneous parts with addition or subtraction, the above cases can be combined.
For any cases not covered, start with the same form as the inhomogeneous part has (for trigonometric functions, duplicate it with both $\sin$ and $\cos$).

\rmvspace\shade{gray}{Other methods}\rmvspace
\begin{itemize}[noitemsep]
    \item \bi{Change of variable} Apply substitution method here, substituting for example for $y' = f(ax + by + c)$ $u = ax + by$ to make the integral simpler.
          Mostly intuition-based (as is the case with integration by substitution)
    \item \bi{Separation of variables} For equations of form $y' = a(y) \cdot b(x)$ (Note: Not linear),
          we transform into $\frac{y'}{a(y)} = b(x)$ and then integrate by substituting $y'(x) dx = dy$, changing the variable of integration.
          Solution: $A(y) = B(x) + c$, with $A = \int \frac{1}{a}$ and $B(x) = \int b(x)$.
          To get final solution, solve the above equation for $y$.
\end{itemize}
