[Analysis] Add some notes

semester3/analysis-ii/cheat-sheet-jh/parts/diffeq/linear-ode/02_constant-coefficient.tex

@@ -5,11 +5,14 @@ The coefficients $a_i$ are constant functions of form $a_i(x) = k$ with $k$ cons
 \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 $\{ z_j \cdot x^{v_j - 1} e^{\gamma_i x} \divides v_j \in \N, \gamma_i \in \R \}$ where $v_j$ is the multiplicity of the root $\gamma_i$.
-          For $\gamma_i = \alpha + \beta i \in \C$, we have $z_1 \cdot e^{\alpha x}\cos(\beta x)$, $z_2 \cdot e^{\alpha x}\sin(\beta x)$, representing the two complex conjugated solutions.
+    \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 $e^{\alpha x}$: