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118 lines
4.1 KiB
TeX
118 lines
4.1 KiB
TeX
\definition \textbf{Differential Equation} (DE)\\
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Equation relating unknown $f$ to derivatives $f^{(i)}$ at \textit{same} $x$.
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\definition \textbf{Ordinary Differential Equation} (ODE)\\
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DE s.t. $f: I \to \R$ is in one variable.
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\definition \textbf{Partial Differential Equation} (PDE)\\
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DE s.t. $f: I^d \to \R$ is in multiple variables.
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\notation $f^{(i)}$ or $y^{(i)}$ instead of $f^{(i)}(x)$ for brevity.
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\definition \textbf{Order} $\ \ord(F) := \underset{i \geq 0}{\text{max}}\{ i \sep f^{(i)} \in F,\ f^{(i)} \neq 0 \}$
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\remark Any $F$ s.t. $\ord(F) \geq 2$ can be reduced to $\ord(F') = 1$, but using functions of higher dimensions.
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\begin{subbox}{Solutions to ODEs}
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\smalltext{$\forall F: \R^2 \to \R$ s.t. $F$ is cont. diff. and $x_0,y_0 \in \R$:}
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\begin{align*}
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& \exists f: I \to \R \\
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& \text{s.t. } \forall x \in I: f'(x) = F(x, f(x)) \text{ and } f(x_0) = y_0
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\end{align*}
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\smalltext{s.t. $I$ is open and maximal.}
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\end{subbox}
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\subtext{Intuition: Solutions always exist (locally!) for \textit{nice enough} equations.}
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\subsection{Linear Differential Equations}
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\definition \textbf{Linear Differential Equation} (LDE)\\
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$$
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y^{(k)} + a_{k-1}y^{(k-1)} + \ldots + a_1y' + a_0y = b
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$$
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\subtext{
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$I \subset \R$ is open$,\quad k \geq 1,\quad \forall i < k: a_i: I \to \C$
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}
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\definition Homogeneity of LDEs\\
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\begin{tabular}{ll}
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\textbf{Homogeneous} & $\iffdef b = 0$\\
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\textbf{Inhomogeneous} & $\iffdef b \neq 0$
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\end{tabular}
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\remark $D(y) := y^{(k)} + \ldots + a_0y$ is a linear operation:
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$$
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D(z_1f_1 + z_2f_2) = z_1D(f_1) + z_2D(f_2)
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$$
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\subtext{
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$\forall z_1,z_2 \in \C,\quad f_1,f_2\ k$-times differentiable:
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}
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\definition \textbf{Homogeneous Solution Space}\\
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$\S(F) := \{ f: I \to \C \sep f \text{ solves } F, f \text{ is } k \text{-times diff.} \}$
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\remark $\S(F)$ is the Nullspace of a lin. map: $f$ to $D(f)$:
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$$
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D(f) = z_1D(f_1) + z_2D(f_2) = 0
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$$
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\subtext{ $\forall z_1,z_2 \in \C,\quad f_1,f_2 \in \S$ }
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\begin{subbox}{Solutions for complex homogeneous LDEs}
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\smalltext{ $F$ s.t. $a_0,\ \ldots\ ,a_{k-1}$ continuous and complex-valued }
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\begin{enumerate}
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\item $\S$ is a complex vector space, $\dim(\S) = k$
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\item $\S$ is a subspace of $\{ f \sep f: I \to \C \}$
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\item $\forall x_0 \in I, (y_0,\ldots,y_{k-1}) \in \C^k$ a unique sol. exists
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\end{enumerate}
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\end{subbox}
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\begin{subbox}{Solutions for real homogeneous LDEs}
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\smalltext{$F$ s.t. $a_0,\ \ldots\ ,a_{k-1}$ continuous and real-valued}
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\begin{enumerate}
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\item $\S$ is a real vector space, $\dim(\S) = k$
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\item $\S$ is a subspace of $\{ f \sep f: I \to \R \}$
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\item $\forall x_0 \in I, (y_0,\ldots,y_{k-1}) \in \R^k$ a unique sol. exists
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\end{enumerate}
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\end{subbox}
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\definition \textbf{Inhomogeneous Solution Space}\\
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$\S_b(F) := \{ f + f_0 \sep f \in \S(F),\ f_0 \text{ is a particular sol.} \}$\\
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\subtext{Note: This is only a vector space if $b = 0$, where $\S_b = \S$.}
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\begin{subbox}{Solutions for real inhomogeneous LDEs}
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\smalltext{$F$ s.t. $a_0,\ \ldots\ ,a_{k-1}$ continuous, $b: I \to \C$}
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\begin{enumerate}
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\item $\forall x_0 \in I, (y_0,\ldots,y_{k-1}) \in \C^k$ a unique sol. exists
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\item If $b, a_i$ are real-valued, a real-valued sol. exists.
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\end{enumerate}
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\end{subbox}
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\remark \textbf{Applications of Linearity}\\
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If $f_1$ solves $F$ for $b_1$, and $f_2$ for $b_2$: $f_1 + f_2$ solves $b_1 + b_2$. \\
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Follows from: $D(f_1) + D(f_2) = b_1 + b_2$.
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\newpage
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\subsection{Finding Solutions: First Order}
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\subtext{ $I \subset \R, \quad a,b: I \to \R$ }
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$$ y' + ay = b $$
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Approach:
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\begin{enumerate}
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\item Hom. Solution: $y' + ay = 0$ using $f_1 = ke^{-A(x)}$\\
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\subtext{Note that $\S$ has $\dim(\S) = 1$, so $f_1 \neq 0$ is a Basis for $\S$}
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\item Part. Solution: $f_0 \in \S_b$ using Variance of Parameters
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\end{enumerate}
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Solutions: $ f_0 + zf_1 \quad \text{ for } z \in \C $
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\begin{subbox}{Explicit Solution for 1st Order LDEs}
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\smalltext{$A(x)$ is a primitive of $a$, $f(x_0) = y_0$}
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\begin{align*}
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f(x) &= z \cdot \exp(-A(x)) \\
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f(x) &= y_0 \cdot \exp(A(x_0) - a(x))
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\end{align*}
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\end{subbox}
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