diff --git a/semester3/analysis-ii/analysis-ii-cheat-sheet.pdf b/semester3/analysis-ii/analysis-ii-cheat-sheet.pdf index 33fbac9..d5f29da 100644 Binary files a/semester3/analysis-ii/analysis-ii-cheat-sheet.pdf and b/semester3/analysis-ii/analysis-ii-cheat-sheet.pdf differ diff --git a/semester3/analysis-ii/parts/diffeq/linear-ode/01_order-one.tex b/semester3/analysis-ii/parts/diffeq/linear-ode/01_order-one.tex index 1e62244..7689bdb 100644 --- a/semester3/analysis-ii/parts/diffeq/linear-ode/01_order-one.tex +++ b/semester3/analysis-ii/parts/diffeq/linear-ode/01_order-one.tex @@ -5,3 +5,5 @@ \bi{(3)} If initial conditions, find equations $\in \mathcal{S}_b$ which fulfill conditions using SLE (as always) \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.