diff --git a/semester3/analysis-ii/cheat-sheet-rb/main.pdf b/semester3/analysis-ii/cheat-sheet-rb/main.pdf index 64f6c63..565d8f8 100644 Binary files a/semester3/analysis-ii/cheat-sheet-rb/main.pdf and b/semester3/analysis-ii/cheat-sheet-rb/main.pdf differ diff --git a/semester3/analysis-ii/cheat-sheet-rb/main.tex b/semester3/analysis-ii/cheat-sheet-rb/main.tex index b4b19ee..9571fdd 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/main.tex +++ b/semester3/analysis-ii/cheat-sheet-rb/main.tex @@ -11,5 +11,9 @@ \section{Differential Equations} \input{parts/01_diffeq.tex} + +\newpage +\section{Differential Calculus in $\R^n$} +\input{parts/02_diff.tex} \end{document} diff --git a/semester3/analysis-ii/cheat-sheet-rb/parts/01_diffeq.tex b/semester3/analysis-ii/cheat-sheet-rb/parts/01_diffeq.tex index 99526d2..1370abe 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/parts/01_diffeq.tex +++ b/semester3/analysis-ii/cheat-sheet-rb/parts/01_diffeq.tex @@ -26,12 +26,8 @@ DE s.t. $f: I^d \to \R$ is in multiple variables. \subsection{Linear Differential Equations} \definition \textbf{Linear Differential Equation} (LDE)\\ -$$ -y^{(k)} + a_{k-1}y^{(k-1)} + \ldots + a_1y' + a_0y = b -$$ -\subtext{ - $I \subset \R$ is open$,\quad k \geq 1,\quad \forall i < k: a_i: I \to \C$ -} +$$y^{(k)} + a_{k-1}y^{(k-1)} + \ldots + a_1y' + a_0y = b$$ +\subtext{$I \subset \R$ is open$,\quad k \geq 1,\quad \forall i < k: a_i: I \to \C$} \definition Homogeneity of LDEs\\ \begin{tabular}{ll} @@ -40,12 +36,8 @@ $$ \end{tabular} \remark $D(y) := y^{(k)} + \ldots + a_0y$ is a linear operation: -$$ -D(z_1f_1 + z_2f_2) = z_1D(f_1) + z_2D(f_2) -$$ -\subtext{ - $\forall z_1,z_2 \in \C,\quad f_1,f_2\ k$-times differentiable: -} +$$D(z_1f_1 + z_2f_2) = z_1D(f_1) + z_2D(f_2)$$ +\subtext{$\forall z_1,z_2 \in \C,\quad f_1,f_2\ k$-times differentiable} \definition \textbf{Homogeneous Solution Space}\\ $\S(F) := \{ f: I \to \C \sep f \text{ solves } F, f \text{ is } k \text{-times diff.} \}$ @@ -94,24 +86,122 @@ If $f_1$ solves $F$ for $b_1$, and $f_2$ for $b_2$: $f_1 + f_2$ solves $b_1 + b_ Follows from: $D(f_1) + D(f_2) = b_1 + b_2$. \newpage -\subsection{Finding Solutions: First Order} +\subsection{Linear Solutions: First Order} \subtext{ $I \subset \R, \quad a,b: I \to \R$ } -$$ y' + ay = b $$ -Approach: -\begin{enumerate} - \item Hom. Solution: $y' + ay = 0$ using $f_1 = ke^{-A(x)}$\\ - \subtext{Note that $\S$ has $\dim(\S) = 1$, so $f_1 \neq 0$ is a Basis for $\S$} - \item Part. Solution: $f_0 \in \S_b$ using Variance of Parameters -\end{enumerate} -Solutions: $ f_0 + zf_1 \quad \text{ for } z \in \C $ -\begin{subbox}{Explicit Solution for 1st Order LDEs} +\textbf{Form:} +$$ y' + ay = b $$ +\textbf{Approach:} +\begin{enumerate} + \item Hom. Solution $f_1$ for: $y' + ay = 0$\\ + \subtext{Note that $\S$ has $\dim(\S) = 1$, so $f_1 \neq 0$ is a Basis for $\S$} + \item Part. Solution $f_0$ for $y' + ay = b$ +\end{enumerate} +\textbf{Solutions:} $ f_0 + zf_1 \quad \text{ for } z \in \C $ + +\begin{subbox}{Explicit Homogeneous Solution} \smalltext{$A(x)$ is a primitive of $a$, $f(x_0) = y_0$} \begin{align*} - f(x) &= z \cdot \exp(-A(x)) \\ - f(x) &= y_0 \cdot \exp(A(x_0) - a(x)) + f_1(x) &= z \cdot \exp(-A(x)) \\ + f_1(x) &= y_0 \cdot \exp(A(x_0) - a(x)) \end{align*} \end{subbox} +Variation of Constants: Treating $z$ as $z(x)$ yields: +\begin{subbox}{Explicit Inhomogeneous Solution} + \smalltext{$A(x)$ is a primitive of $a$} + $$ + f_0(x) = \underbrace{\left(\int b(x)\cdot\exp(A(x)) \right)}_{z(x)} \cdot \exp\left(-A(x)\right) + $$ +\end{subbox} +\method \textbf{Educated Guess}\\ +Usually, $y$ has a similar form to $b$: + +\begin{tabular}{ll} + \hline + $b(x)$ & \text{Guess} \\ + \hline + $a \cdot e^{\alpha x}$ & $b \cdot e^{\alpha x}$ \\ + $a \cdot \sin(\beta x)$ & $c\sin(\beta x) + d\cos(\beta x)$\\ + $b \cdot \cos(\beta x)$ & $c\sin(\beta x) + d\cos(\beta x)$\\ + $ae^{\alpha x} \cdot \sin(\beta x)$ & $e^{\alpha x}\left(c\sin(\beta x) + d\cos(\beta x)\right)$\\ + $be^{\alpha x} \cdot \cos(\beta x)$ & $e^{\alpha x}\left(c\sin(\beta x) + d\cos(\beta x)\right)$\\ + $P_n(x) \cdot e^{\alpha x}$ & $R_n(x) \cdot e^{\alpha x}$\\ + $P_n(x) \cdot e^{\alpha x}\sin(\beta x)$ & $e^{\alpha x}\left( R_n(x) \sin(\beta x) + S_n(x) \cos(\beta x) \right)$\\ + $P_n(x) \cdot e^{\alpha x}\cos(\beta x)$ & $e^{\alpha x}\left( R_n(x) \sin(\beta x) + S_n(x) \cos(\beta x) \right)$\\ + \hline +\end{tabular} + +\remark If $\alpha, \beta$ are roots of $P(X)$ with multiplicity $j$, multiply guess with a $P_j(x)$. + +\subsection{Linear Solutions: Constant Coefficients} +\textbf{Form:} +$$ + y^{(k)} + a_{k-1}y^{(k-1)} + \ldots + a_1y' + a_0y = b +$$ +\subtext{Where $a_0, \ldots, a_{k-1} \in \C$ are constants, $b(x)$ is continuous.} + +\subsubsection{Homogeneous Equations} + +The idea is to find a Basis of $\S$: + +\definition \textbf{Characteristic Polynomial} $P(X) = \prod_{i=1}^{k} (X-\alpha_i)$ + +\remark The unique roots $\alpha_1,\ldots,\alpha_l$ form a Basis: +$$ + \text{span}(\S) = \{ x^je^{\alpha_i x} \sep i \leq l,\quad 0 \leq j \leq v_i \} +$$ +\subtext{$v_1,\ldots,v_k$ are the Multiplicities of $\alpha_1,\ldots,\alpha_k$} + +\remark If $\alpha_j = \beta + \gamma i \in \C$ is a root, $\bar{\alpha_j} = \beta - \gamma i$ is too.\\ +To get a real-valued solution, apply: +$$ +e^{\alpha_j x} = e^{\beta x}\left( \cos(\gamma x) + i \sin(\gamma x) \right) +$$ + +\begin{subbox}{Explicit Homogeneous Solution} + \smalltext{Using $\alpha_1,\ldots,\alpha_k$ from $P(X)$ s.t. $\alpha_i \neq \alpha_j$, $z_i \in \C$ arbitrary} + $$ + f(x) = \prod_{i=1}^{k} z_i \cdot e^{\alpha_i x} \quad\text{with}\quad f^{(j)(x)} = \prod_{i=1}^{k} z_i \cdot \alpha_i^j e^{\alpha_i x} + $$ + \smalltext{Multiple roots: same scheme, using the basis vectors of $\S$} +\end{subbox} +\subtext{Solutions exist $\forall Z = (z_1,\ldots,z_k)$ since that system's $\det(M_Z) \neq 0$.} + +\newpage +\subsubsection{Inhomogeneous Equations} + +\method \textbf{Undetermined Coefficients}: An educated guess. +\begin{enumerate} + \item $b(x) = cx^d \cdot e^{\alpha x} \implies f_p(x) = Q(x)e^{\alpha x}$\\ + \subtext{$\deg(Q) \leq d + v_\alpha$, where $v_\alpha$ is $\alpha$'s multiplicity in $P(X)$} + \item $\begin{rcases*} + b(x) = cx^d \cdot \cos(\alpha x) \\ + b(x) = cx^d \cdot \sin(\alpha x) + \end{rcases*} f_p = Q_1(x)\cos(\alpha x) + Q_2(x)\sin(\alpha x)$ + \subtext{$\deg(Q_{1,2}) \leq d + v_\alpha$, where $v_\alpha$ is $\alpha$'s multiplicity in $P(X)$} +\end{enumerate} + +\remark \textbf{Applying Linearity}\\ +If $b(x) = \sum_{i=1}^{n} b_i(x)$, A solution for $b(x)$ is $f(x) = \sum_{i=1}^{n} f_i(x)$\\ +\subtext{Sometimes called \textit{Superposition Principle} in this context} + +\subsection{Other Methods} + +\method \textbf{Change of Variable}\\ +If $f(x)$ is replaced by $h(y) = f(g(y))$, then $h$ is a sol. too.\\ +\subtext{Changes like $h(t) = f(e^t)$ may lead to useful properties.} + +\begin{subbox}{Separation of Variables} + Form: + $$ + y' = a(y)\cdot b(x) + $$ + Solve using: + $$ + \int \frac{1}{a(y)}\ \text{d}y = \int b(x) \dx + c + $$ +\end{subbox} +\subtext{Usually $\int 1/a(y)\ \text{d}y$ can be solved directly for $\ln|a(y)|+c$.} \ No newline at end of file diff --git a/semester3/analysis-ii/cheat-sheet-rb/parts/02_diff.tex b/semester3/analysis-ii/cheat-sheet-rb/parts/02_diff.tex new file mode 100644 index 0000000..e69de29 diff --git a/semester3/analysis-ii/cheat-sheet-rb/util/helpers.tex b/semester3/analysis-ii/cheat-sheet-rb/util/helpers.tex index a98c347..02d2dbc 100644 --- a/semester3/analysis-ii/cheat-sheet-rb/util/helpers.tex +++ b/semester3/analysis-ii/cheat-sheet-rb/util/helpers.tex @@ -52,6 +52,7 @@ \def \notation{\colorbox{lightgray}{Notation} } \def \remark{\colorbox{lightgray}{Remark} } \def \theorem{\colorbox{lightgray}{Th.} } +\def \method{\colorbox{lightgray}{Method} } % For intuiton and less important notes \def \subtext#1{