[Analysis] Various fixes and optimizations

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

@@ -16,11 +16,18 @@ 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
+          $f_p = Q(x) e^{\alpha x}$ 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 $b(x) = c x^d \sin(\alpha x)$:
-          $f_p = Q_1(x) \cdot \cos(\alpha x) + Q_2(x9 \cdot \sin(\alpha x))$,
-          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
+          $f_p = Q_1(x) \cdot \cos(\alpha x) + Q_2(x) \cdot \sin(\alpha x))$,
+          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
+          $Q_1(x) e^{\alpha x} \sin(\beta x) + Q_2(x) e^{\alpha x} \cos(\beta x)$, agasin with the same polynomial.
+          Often, it is sufficent to have a polynomial of degree 0 (i.e. constant)
 \end{enumerate}
+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]