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[Analysis] Improve order 1 ode and constant coefficient summary

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Janis Hutz
2025-10-08 14:24:23 +02:00
parent c391656c30
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4 changed files with 17 additions and 8 deletions
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semester3/analysis-ii/parts/diffeq/linear-ode/01_order-one.tex
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@@ -1,8 +1,6 @@
\newsectionNoPB
\subsection{Linear differential equations of first order}
\shade{gray}{Finding solution set} \bi{(1)} Find basis $\{ f_1, \ldots, f_k \}$ for $\mathcal{S}_0$ for homogeneous equation (set $b(x) = 0$).
\bi{(2)} If inhom. find $f_p$ that solves the equation. The set of solutions $\mathcal{S}_b = \{ f_h + f_p \divides f_h \in \mathcal{S_0} \}$.
\bi{(3)} If initial conditions, find equations $\in \mathcal{S}_b$ which fulfill conditions using SLE (as always)
\shade{gray}{Homogeneous equation} Move all $y$ to one side and all other vars to other. Integrate both
\shortproposition Solution of $y' + ay = 0$ is of form $f(x) = z e^{-A(x)}$ with $A$ anti-derivative of $a$