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a/semester3/analysis-ii/cheat-sheet-jh/parts/diffeq/linear-ode/00_intro.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/diffeq/linear-ode/00_intro.tex index bc43a83..a082be9 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/diffeq/linear-ode/00_intro.tex +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/diffeq/linear-ode/00_intro.tex @@ -19,7 +19,7 @@ The solution space $\mathcal{S}$ is spanned by $k$ functions, which thus form a \shade{gray}{Finding solutions (in general)} \rmvspace \begin{enumerate}[label=\bi{(\arabic*)}, noitemsep] - \item Find basis $\{ f_1, \ldots, f_k \}$ for $\mathcal{S}_0$ for homogeneous equation (set $b(x) = 0$) (i.e. find homogeneous part, solve it) - \item If inhomogeneous, find $f_p$ that solves the equation. The set of solutions is then $\mathcal{S}_b = \{ f_h + f_p \divides f_h \in \mathcal{S}_0 \}$. - \item If there are initial conditions, find equations $\in \mathcal{S}_b$ which fulfill conditions using SLE (as always) + \item Find the solution to the homogeneous equation ($b(x) = 0$) using one of the methods below + \item If inhomogeneous, use a method below for an Ansatz for $f_p$, derive it and input that into the diffeq and solve. + \item If there are initial conditions, find equations $\in \cS$ which fulfill conditions using SLE (as always) \end{enumerate} diff --git 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 index 54c10ee..2c7c749 100644 --- 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 @@ -7,6 +7,6 @@ \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 $C$ + \item Find solution to $y_p = z(x) e^{-A(x)}$ with $z(x) = \int b(x) e^{A(x)} \dx x$ ($A(x)$ in exp here!), + \item Solve and the result is $y(x) = y_h + c \cdot y_p$. For initial value problem, determine coefficient $C$ \end{enumerate}