From f0c472bb878c8965ecbc12e85b3878a1ffa4c321 Mon Sep 17 00:00:00 2001 From: RobinB27 Date: Sun, 11 Jan 2026 11:40:21 +0100 Subject: [PATCH] [Analysis] Add example --- semester3/analysis-ii/cheat-sheet-rb/main.pdf | Bin 328650 -> 330786 bytes .../cheat-sheet-rb/parts/03_diffeq_sol.tex | 81 ++++++++++-------- 2 files changed, 46 insertions(+), 35 deletions(-) diff --git a/semester3/analysis-ii/cheat-sheet-rb/main.pdf b/semester3/analysis-ii/cheat-sheet-rb/main.pdf index 53f35c437b7e0754ff6d1b85c60a16558f158722..8f1fed05c72e0721af0bebe1ab11d3c1514eee8c 100644 GIT binary patch delta 110768 zcmV(Gn0XXD1Yr;TX)}2PXd1y;*s~z2|CPdj4nj_kZ}Y zSqPmdZ6*DBsdATyTrZ3=NiJpad-ZTBM4BEvyEwhxZ7$BY50@&{2fK^n?^m0jQlZvY z>+{`aby_@o{LA&KXo^=CZxKmujDJhISW2A;Rdt_;@&$j{oUd08l{mOUFV;0LJbbv+ zS$^>K=4!j6f3gSPT^+Bl>du?_qknHuz~rvz`)lg$;-N|puFejN=F$btl}%Kd;a9%( zP1zz}*d%uNKsMoamJ@Yd9q^ftVZkT?AN~AMYlSfP+ z0t+Z?)#62GWkHU7;V}ACP;;pgm)7t6w&+lcpoLAWuKW8bRkCao_L3zzdVek3d|y&M 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\cos(\beta x)$ & $c\sin(\beta x) + d\cos(\beta x)$\\ - $ae^{\alpha x} \cdot \sin(\beta x)$ & $e^{\alpha x}\left(c\sin(\beta x) + d\cos(\beta x)\right)$\\ - $be^{\alpha x} \cdot \cos(\beta x)$ & $e^{\alpha x}\left(c\sin(\beta x) + d\cos(\beta x)\right)$\\ - $P_n(x) \cdot e^{\alpha x}$ & $R_n(x) \cdot e^{\alpha x}$\\ - $P_n(x) \cdot e^{\alpha x}\sin(\beta x)$ & $e^{\alpha x}\left( R_n(x) \sin(\beta x) + S_n(x) \cos(\beta x) \right)$\\ - $P_n(x) \cdot e^{\alpha x}\cos(\beta x)$ & $e^{\alpha x}\left( R_n(x) \sin(\beta x) + S_n(x) \cos(\beta x) \right)$\\ - \hline -\end{tabular} +\begin{footnotesize} + \begin{center} + \begin{tabular}{ll} + \hline + $b(x)$ & \text{Guess} \\ + \hline + $a \cdot e^{\alpha x}$ & $b \cdot e^{\alpha x}$ \\ + $a \cdot \sin(\beta x)$ & $c\sin(\beta x) + d\cos(\beta x)$\\ + $b \cdot \cos(\beta x)$ & $c\sin(\beta x) + d\cos(\beta x)$\\ + $ae^{\alpha x} \cdot \sin(\beta x)$ & $e^{\alpha x}\left(c\sin(\beta x) + d\cos(\beta x)\right)$\\ + $be^{\alpha x} \cdot \cos(\beta x)$ & $e^{\alpha x}\left(c\sin(\beta x) + d\cos(\beta x)\right)$\\ + $P_n(x) \cdot e^{\alpha x}$ & $R_n(x) \cdot e^{\alpha x}$\\ + $P_n(x) \cdot e^{\alpha x}\sin(\beta x)$ & $e^{\alpha x}\left( R_n(x) \sin(\beta x) + S_n(x) \cos(\beta x) \right)$\\ + $P_n(x) \cdot e^{\alpha x}\cos(\beta x)$ & $e^{\alpha x}\left( R_n(x) \sin(\beta x) + S_n(x) \cos(\beta x) \right)$\\ + \hline + \end{tabular} + \end{center} +\end{footnotesize} \remark If $\alpha, \beta$ are roots of $P(X)$ with multiplicity $j$, multiply guess with a $P_j(x)$. \subsection{Linear Solutions: Constant Coefficients} -\textbf{Form:} -$$ - y^{(k)} + a_{k-1}y^{(k-1)} + \ldots + a_1y' + a_0y = b -$$ +\textbf{Form:} $ y^{(k)} + a_{k-1}y^{(k-1)} + \ldots + a_1y' + a_0y = b$\\ \subtext{Where $a_0, \ldots, a_{k-1} \in \C$ are constants, $b(x)$ is continuous.} \subsubsection{Homogeneous Equations} - -The idea is to find a Basis of $\S$: +\smalltext{The idea is to find a Basis of $\S$:} \definition \textbf{Characteristic Polynomial} $P(X) = \prod_{i=1}^{k} (X-\alpha_i)$ @@ -82,7 +82,6 @@ $$ \end{subbox} \subtext{Solutions exist $\forall Z = (z_1,\ldots,z_k)$ since that system's $\det(M_Z) \neq 0$.} -\newpage \subsubsection{Inhomogeneous Equations} \method \textbf{Undetermined Coefficients}: An educated guess. @@ -98,13 +97,24 @@ $$ \remark \textbf{Applying Linearity}\\ If $b(x) = \sum_{i=1}^{n} b_i(x)$, A solution for $b(x)$ is $f(x) = \sum_{i=1}^{n} f_i(x)$\\ -\subtext{Sometimes called \textit{Superposition Principle} in this context} +\subtext{Sometimes called \textit{Superposition Principle} in this context.} \subsection{Other Methods} \method \textbf{Change of Variable}\\ If $f(x)$ is replaced by $h(y) = f(g(y))$, then $h$ is a sol. too.\\ -\subtext{Changes like $h(t) = f(e^t)$ may lead to useful properties.} +\subtext{Changes like $h(t) = f(e^t)$ may lead to, i.e. ODEs in constant coeffs} + +\begin{footnotesize} + \textbf{Example:} $2xy' - y = 0$ \\ + Using substitution: $x = e^t,\quad h(t) = y(e^t),\quad h'(t) = e^t \cdot y'(e^t)$ + \begin{enumerate} + \item $2x \cdot y'(x) = 2 \cdot h'(t)$ + \item $-y(x)$ = $-h(t)$ + \end{enumerate} + So: $2xy' - y \overset{\text{sub}}{=} 2h'(t) - h(t) = 0$ \\ + Yields: $h(t) = \alpha \cdot e^\frac{t}{2} \overset{\text{resub}}{\implies} y(x) = \alpha \cdot e^\frac{\ln(x)}{2} = \alpha \cdot \sqrt{x}$ +\end{footnotesize} \begin{subbox}{Separation of Variables} Form: @@ -119,15 +129,16 @@ If $f(x)$ is replaced by $h(y) = f(g(y))$, then $h$ is a sol. too.\\ \subtext{Usually $\int 1/a(y)\ \text{d}y$ can be solved directly for $\ln|a(y)|+c$.} \subsection{Method Overview} - -\begin{center} - \begin{tabular}{l|l} - \textbf{Method} & \textbf{Use case} \\ - \hline - Variation of constants & LDE with $\ord(F)=1$ \\ - Characteristic Polynomial & Hom. LDE w/ const. coeff. \\ - Undetermined Coefficients & Inhom. LDE w/ const. coeff. \\ - Separation of Variables & ODE s.t. $y' = a(y)\cdot b(x)$ \\ - Change of Variables & e.g. $y' = f(ax + by + c)$ \\ - \end{tabular} -\end{center} \ No newline at end of file +\begin{footnotesize} + \begin{center} + \begin{tabular}{l|l} + \textbf{Method} & \textbf{Use case} \\ + \hline + Variation of constants & LDE with $\ord(F)=1$ \\ + Characteristic Polynomial & Hom. LDE w/ const. coeff. \\ + Undetermined Coefficients & Inhom. LDE w/ const. coeff. \\ + Separation of Variables & ODE s.t. $y' = a(y)\cdot b(x)$ \\ + Change of Variables & e.g. $y' = f(ax + by + c)$ \\ + \end{tabular} + \end{center} +\end{footnotesize} \ No newline at end of file