From 41b2a483e629c2b99257589abe6f48c73773e57b Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Mon, 6 Oct 2025 13:07:17 +0200 Subject: [PATCH] [Analysis] Start constant coefficients section --- .../analysis-ii/analysis-ii-cheat-sheet.pdf | Bin 206082 -> 209309 bytes .../parts/diffeq/linear-ode/01_order-one.tex | 2 ++ .../linear-ode/02_constant-coefficient.tex | 5 ++++- 3 files changed, 6 insertions(+), 1 deletion(-) diff --git a/semester3/analysis-ii/analysis-ii-cheat-sheet.pdf b/semester3/analysis-ii/analysis-ii-cheat-sheet.pdf index 33fbac9e5898491fda8926cbf33e1109251b6d8b..d5f29daa2967a05cc9a1e4386a2ce6b046b36ef6 100644 GIT binary patch delta 81192 zcmV(vKFt&xzNGQ*KI1=XxT)e7C=tG3=g_M1K<(rP?~jl=q*x&gFCoYPWk5zw-rWr>L5 zRO*DisoMc444xeQQ$#_n6BJ~n3~CWBX{i#f@qXWb%=4cC-vE*>yWLwgg@on}BBG>9 z$OxV=p|qbjb1#7m!w-5s3|3%j`14J2Z(VRN{c#8+W_!j+()0O2qNQpcN+a1Ip|Xh(u-p45FF+yt&%D3IwVK5sPhdB zeUtyr!!7cr&lTfj5R3^fqs7fE@0aj}QqfhX*Bbw(Q@UTR&q(`QwgO%bm*E9jDv}QT z3^A%M{s7HV2yzN#Ze(+Ga%Ev{3T19&Z(_5&3j+uQG%z_ZlVJoXf9+gblN+~{exF~_ z*@vxW#YAv#v|XuMCz4anddsn;H+vsioaT@#4mlz@mXu$==it%}Gy!rb$#TlOCCeny zXy5=im+yQ4M60`sR)4#p{_`&{Z~pX+logXiNX@I4OZ>_jQ5j~4;H-LiTm4k?*_<)1 z>SybpDHHv^-_8W9e~;^KGn

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\shortproposition Solution of $y' + ay = 0$ is of form $f(x) = z e^{-A(x)}$ with $A$ anti-derivative of $a$ + +\TODO Improve procedure with notes from session \& SPAM diff --git a/semester3/analysis-ii/parts/diffeq/linear-ode/02_constant-coefficient.tex b/semester3/analysis-ii/parts/diffeq/linear-ode/02_constant-coefficient.tex index 8b0fa9d..28b48c3 100644 --- a/semester3/analysis-ii/parts/diffeq/linear-ode/02_constant-coefficient.tex +++ b/semester3/analysis-ii/parts/diffeq/linear-ode/02_constant-coefficient.tex @@ -1,3 +1,6 @@ \newsectionNoPB \subsection{Linear differential equations with constant coefficients} -The coefficients $a_i$ are constant functions of form $a_i(x) = k$ with $k$ constant. +The coefficients $a_i$ are constant functions of form $a_i(x) = k$ with $k$ constant, where $b(x)$ can be any function.\\ +% +\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 $a_i$). +Find the roots of polynomial.