[Analysis] LDE

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}

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$$
-y^{(k)} + a_{k-1}y^{(k-1)} + \ldots + a_1y' + a_0y = b
+\subtext{$I \subset \R$ is open$,\quad k \geq 1,\quad \forall i < k: a_i: I \to \C$}
-$$
-\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)$$
-D(z_1f_1 + z_2f_2) = z_1D(f_1) + z_2D(f_2)
+\subtext{$\forall z_1,z_2 \in \C,\quad f_1,f_2\ k$-times differentiable}
-$$
-\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_1(x) &= z \cdot \exp(-A(x)) \\
-        f(x) &= y_0 \cdot \exp(A(x_0) - 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 $\ln|a(y)|+c$.}

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{