From 33423c4698333a68dbb549a0bbc4f16a81a44b4f Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Fri, 30 Jan 2026 16:00:54 +0100 Subject: [PATCH] [Analysis] Add some notes --- .../cheat-sheet-jh/cheat-sheet-de.pdf | Bin 1090106 -> 1090392 bytes .../cheat-sheet-jh/cheat-sheet-en.pdf | Bin 1083366 -> 1082379 bytes .../cheat-sheet-jh/parts/fields.tex | 2 ++ .../analysis-ii-cheat-sheet.pdf | Bin 350263 -> 350136 bytes .../parts/diffeq/linear-ode/01_order-one.tex | 4 ++-- .../linear-ode/02_constant-coefficient.tex | 9 ++++++--- 6 files changed, 10 insertions(+), 5 deletions(-) diff --git a/semester2/analysis-i/cheat-sheet-jh/cheat-sheet-de.pdf b/semester2/analysis-i/cheat-sheet-jh/cheat-sheet-de.pdf index 6297d15e6e468a9d6a279d10da7d7b03a55c891c..293c04bd7463ece0ae1f9c49e5fe94a7d1962a95 100644 GIT binary patch delta 113528 zcmV)MK)AoU#6sA{La!{( zE${&2Sa43d77&TER0I*jG3Bm>*Lm4hCrS9SZO&~y=y9VSv>hYpF~T$!%+*^q%}M>} z;lsa8{Uy|lXVq$`R~Q1KIEaXdiLiAkifxb3`Gllln``@@Ad0h?a3tSY*JMYPHX=F zU|)=a1UYggGMxUl^m=0m^|r+vLQnH{W^8F``0}g8;Eh%dfY7e&-_MAnH(+Z}>eR9- zF!=egEhI~L49v?=Hj>_T0K<1Ay(d+1o|jd0iTp^GjC^&|=94&-f1aD!_mT!sQSNHP zpp?cO!a@@UQMMHZ1sk_a4CJciZCy(ZLDYu%{lMcX6`oOhOd2!^l7j!~LL~h*r(tao9V(#WrI)NFz%GTQW$c41zE+kTa z8E=;k>z2&nbZ9~btrk^Ttz@G+5tD(483G7`|LaWij0SZDP`yD32h=;|Ffp0Yxh&L7 zRW10U8lHzXe;rD%MP-h#+_)94!+8Y726hiGC0<=*zY*5+5uICrY?rJLvss^K>NCPr z8YGp}(_HVI0Lpew0I@y+^mtlG)yPiP&=LH5 z{KlD4w+mPBplcF?WM=Q}vItLCMeXia9&(`^aPI>31%-5oF8HdLwC%Q7X z&eSS*QR{6bpne%}#8AY;`n=HInbb2?JL-HXixa4R<00pGl;lUyyi$gxejCl61y*vE zm!;KSf3He&mbj4%oflSlaiHduc;yB9dr*3zn3HUK0r6gXJ(D2xSgEXi8nbma%}jsB zF`NFTs!)yf%QTGzaXa$#GSyS>m+Ss`ix~JpWgEZV%O(To#{kM!~BaSzQ4%d z&w%DbJ%+`I+}#0CY$i)`ZREgIyRQRGo)U+;nBcJcwXmsBkaH;+jqqn-xXYGEVq@_` z^X*-3j+K+peI^TxdTL_w%0=i~SxuUGFdYxY4w`(mI@%1z5JliuyVJP+C)O8Ycaou4 zq~_QNF*#W5KNH@pvUm94AC?6-gOd{wNC7mHVFW0Xgb)w{CpD9n5Ga35GQ51}en?7{ zB9vI77$=OvKQ0M2mC%G)!l{IU5w5HvFeDh~gc~BX%7~yuatocDNXukTNTP&5hH(uA%;U;6~rLlM#Dc&dj^Iq1!*bNLJ`ZLnHd5iSO!8^?1Dws z63Z#n8L@&IlEILWoMg}?4I`O`sZ{rHsnQ0N2AG@p^R1jx;phKAHV#(<~<^)pVP0eWE^$>J(7LZ(r16;~uP0(&epG)@NO z42=lASthZ8tWc1Hp*blm?lD|TnNftv%HFsDEXAk*xtT-`C5LrjE#O|@XeEcn6)1=? 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a/semester3/analysis-ii/cheat-sheet-jh/parts/diffeq/linear-ode/01_order-one.tex +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/diffeq/linear-ode/01_order-one.tex @@ -1,12 +1,12 @@ \newsectionNoPB \subsection{Linear differential equations of first order} \rmvspace -\shortproposition Solution of $y' + ay = 0$ is of form $f(x) = z e^{-A(x)}$ with $A$ anti-derivative of $a$ +\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} \rmvspace \begin{enumerate}[noitemsep] \item Plug all values into $y_p = \int b(x) e^{A(x)}$ ($A(x)$ in the exponent instead of $-A(x)$ as in the homogeneous solution) - \item Solve and the final $y(x) = y_h + y_p$. For initial value problem, determine coefficient $z$ + \item Solve and the final $y(x) = y_h + y_p$. For initial value problem, determine coefficient $C$ \end{enumerate} diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/diffeq/linear-ode/02_constant-coefficient.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/diffeq/linear-ode/02_constant-coefficient.tex index 1ddf1a4..46b9b0a 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/diffeq/linear-ode/02_constant-coefficient.tex +++ b/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 $e^{\alpha x}$: