[Analysis] More on constant coeff section

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

@@ -2,5 +2,6 @@
 \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.\\
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-\shade{gray}{Solving:} \bi{(1)} Find \textit{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 $a_i$).
+\shade{gray}{Homo. Sol.} Find \textit{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 $a_i$).
-Find the roots of polynomial. 
+Find the roots of polynomial. The solution space is given by $\{ x^{v_j} e^{\gamma_i x} \divides v_j \in \N, \gamma_i \in \R \}$ where $v_j$ is the multiplicity of the root $\gamma_i$.
+For $\gamma_i = \alpha + \beta i \in \C$, we have $e^{\alpha x}\cos(\beta x)$, $e^{\alpha x}\sin(\beta x)$.