[Analysis] Add more notes

semester3/analysis-ii/cheat-sheet-jh/parts/diffeq/linear-ode/00_intro.tex

@@ -20,6 +20,6 @@ The solution space $\mathcal{S}$ is spanned by $k$ functions, which thus form a
 \rmvspace
 \begin{enumerate}[label=\bi{(\arabic*)}, noitemsep]
     \item Find the solution to the homogeneous equation ($b(x) = 0$) using one of the methods below
-    \item If inhomogeneous, use a method below for an Ansatz for $f_p$, derive it and input that into the diffeq and solve.
+    \item If inhomogeneous, use a method below for an Ansatz for $f_p$, derive it and input that into the full diffeq and solve.
     \item If there are initial conditions, find equations $\in \cS$ which fulfill conditions using SLE (as always)
 \end{enumerate}

semester3/analysis-ii/cheat-sheet-jh/parts/diffeq/linear-ode/01_order-one.tex

@@ -4,9 +4,9 @@
 \shortproposition Solution of $y' + ay = 0$ is of form $f(x) = C e^{-A(x)}$ with $A$ anti-derivative of $a$
 
 \rmvspace
-\shade{gray}{Imhomogeneous equation} 
+\shade{gray}{Imhomogeneous equation} $y' + ay = b$ with $b$ any function.
 \rmvspace
 \begin{enumerate}[noitemsep]
-    \item Find solution to $y_p = z(x) e^{-A(x)}$ with $z(x) = \int b(x) e^{A(x)} \dx x$ ($A(x)$ in exp here!),  
+    \item Compute $y_p = z(x) e^{-A(x)}$ with $z(x) = \int b(x) e^{A(x)} \dx x$ ($A(x)$ in exp here!),  
     \item Solve and the result is $y(x) = y_h + c \cdot y_p$. For initial value problem, determine coefficient $C$
 \end{enumerate}

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

@@ -16,16 +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 $\deg(Q) \leq j + d$, 
+          $f_p = Q(x) e^{\alpha x}$ 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
-          $Q_1(x) e^{\alpha x} \sin(\beta x) + Q_2(x) e^{\alpha x} \cos(\beta x)$, agasin with the same 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 $\cos$).