From a4057473aac0276e1730c7c78997f08b4e9823a2 Mon Sep 17 00:00:00 2001 From: Robert Date: Thu, 1 Apr 2021 12:02:56 +0200 Subject: [PATCH] solution_methods.tex --- chapters/sections/solution_methods.tex | 272 +++++++++++++++++++++++++ script.pdf | Bin 1112109 -> 1129939 bytes 2 files changed, 272 insertions(+) diff --git a/chapters/sections/solution_methods.tex b/chapters/sections/solution_methods.tex index dcac273..bbeffcd 100644 --- a/chapters/sections/solution_methods.tex +++ b/chapters/sections/solution_methods.tex @@ -134,4 +134,276 @@ F(y(x)) = \int_{x_0}^x g(t) \dd{t} \] \end{thm} +\begin{proof} + $f$ does not have any roots, thus w.l.o.g. $f > 0$. + \begin{equation} + F'(y) = \rec{f(y)} > 0 \implies F \text{ strictly monotonically increasing} + \end{equation} + Therefore there exists an inverse function $H: F(I) \rightarrow I$. According to the theorem about inverse functions, $H$ is $C^1$ and + \begin{equation} + H'(z) = \rec{F'(H(z))} ~~\forall z \in F(I) + \end{equation} + $F(I)$ is an open interval containing the $0$. Then we have + \begin{equation} + y(x) = H(G(x)) ~~x \in I_2 + \end{equation} + where + \begin{equation} + G(x) = \int_{x_0}^x g(t) \dd{t} + \end{equation} + Now choose $I_2$ such that $x_0 \in I_2$ and $G(I_2) \subset F(I)$. Then + \begin{equation} + \begin{split} + y'(x) &= H'(G(x)) \cdot G'(x) \\ + &= \rec{F'(H(G(x)))} \cdot G'(x) \\ + &= \rec{F'(y(x))} \cdot G'(x) \\ + &= f(y(x)) g(x) + \end{split} + \end{equation} + So $y$ solves the ODE. However, if $\tilde{y}: I \rightarrow \realn$ some solution of the IVP, then $\forall x \in I_2$ + \begin{equation} + \begin{split} + G(x) = \int_{x_0}^x g(x) \dd{x} = \int_{x_0}^x \frac{\tilde{y}(x)}{f(\tilde{y}(x))} \dd{x} = \int_{\tilde{y}(x_0)}^{\tilde{y}(x)} \rec{f(y)} \dd{y} = F(\tilde{y}(x)) + \end{split} + \end{equation} + So $\tilde{y}(x) = H(G(x))$ +\end{proof} + +\begin{rem} + $I_2$ is obviously not unique. We can find the biggest possible domain with + \[ + \bigcup_{\substack{x \in I_2 \\ I_2 \text{ open} \\ G(I_2) \subset F(I)}} I_2 = I_{2, \max} + \] +\end{rem} + +\begin{thm} + Let $f: \realn \rightarrow \realn$ be a continuous function, $a, b, c \in \realn$ and $I$ an open interval. + Then $y: I \rightarrow \realn$ is a solution of the ODE + \[ + y' = f(ax + by + c) + \] + if and only if $ u(x) := ax + by + c$ is a solution of + \[ + u' = a + bf(u) + \] +\end{thm} +\begin{hproof} + Consider + \[ + u'(x) = a + by'(x) + \] +\end{hproof} + +\begin{eg}[Euler Homogeneous ODE] + Let $f: \realn \rightarrow \realn$ be a function and $I$ an open interval not containing the $0$. + Then $y: I \rightarrow \realn$ is a solution of the ODE + \[ + y' = f(\frac{y}{x}) + \] + if and only if + \[ + u(x) = \frac{y(x)}{x} + \] + solves the ODE + \[ + u' = \frac{f(u) - u}{x} + \] +\end{eg} + +\begin{eg} + Let $f: \realn \rightarrow \realn$ be continuous and $a_1, a_2, b_1, b_2, c_1, c_2 \in \realn$ such that + \[ + \begin{vmatrix} + a_1 && b_1 \\ + a_2 && b_2 + \end{vmatrix} \ne 0 + \] + Now let $\tilde{x}, \tilde{y}$ be the solutions of the equation system + \begin{align*} + a_1\tilde{x} + b_1\tilde{y} + c_1 &= 0 \\ + a_2\tilde{x} + b_2\tilde{y} + c_2 &= 0 + \end{align*} + Let $I$ be an open interval not containing the $0$. Then $y: I \rightarrow \realn$ is a solution to + \[ + y' = f\left(\frac{a_1x + b_1y + c_1}{a_2x + b_2y + c_2}\right) + \] + if and only if + \begin{align*} + u: I - \tilde{x} &\longrightarrow \realn \\ + x &\longmapsto y(x + \tilde{x}) - \tilde{y} + \end{align*} + is a solution to + \[ + u' = f\left(\frac{a_1 + b_y \frac{u}{x}}{a_2 + b_2 \frac{u}{x}}\right) + \] +\end{eg} +\begin{proof} + Let $y: I \rightarrow \realn$ be a solution to the initial equation. Then + \begin{equation} + \begin{split} + u'(x) &= y'(x + \tilde{x}) = f\left( \frac{a_1(x + \tilde{x}) + b_1 y(x + \tilde{x}) + c_1}{a_2(x + \tilde{x}) + b_2y(x + \tilde{x}) + c_2} \right) \\ + &= f\left( \frac{a_1x + b_1u(x) + a_1\tilde{x} + b_1\tilde{y} + c_1}{a_2x + b_2u(x) + a_2\tilde{x} + b_2\tilde{y} + c_2} \right) \\ + &= f\left(\frac{a_1 + b_1 \frac{u(x)}{x}}{a_2 + b_2 \frac{u(x)}{x}}\right) + \end{split} + \end{equation} + The other direction is left to the reader. +\end{proof} + +\begin{defi}[Exact ODE] + Let $D \subset \realn^2$ be open, and $p, q: D \rightarrow \realn$ continuous. + The ODE + \[ + p(x, y) + q(x, y) y' = 0 + \] + is said to be exact if there exists a $C^1$-function $H: D \rightarrow \realn$, such that + \begin{align*} + \partial_1 H = p && \partial_2 H = q + \end{align*} + Such a function is called a potential function. +\end{defi} + +\begin{thm} + Let $D \subset \realn^2$ be open, and $p, q: D \rightarrow \realn$ continuous. Let + \[ + p(x, y) + q(x, y)y' = 0 + \] + be exact and $H$ a potential function. Furthermore let $I$ be an open interval and $y: I \rightarrow \realn$ a $C^1$-function such that + \[ + \set[x \in I]{(x, y(x))} \subset D + \] + Then $y$ solves the ODE if and only if $\exists c \in \realn$ such that + \[ + H(x, y(x)) = c + \] +\end{thm} +\begin{proof} + \begin{equation} + \begin{split} + \dv{x} H(x, y(x)) &= \partial_1 H(x, y(x)) + \partial_2 H(x, y(x)) y'(x) \\ + &= p(x, y) + q(x, y)y'(x) + \end{split} + \end{equation} +\end{proof} + +\begin{thm} + Let $D \subset \realn^2$ be open, and $p, q: D \rightarrow \realn$ continuously differentiable. If + \[ + p(x, y) + q(x, y)y' = 0 + \] + is exact, then + \[ + \partial_2 p = \partial_1 q + \] +\end{thm} +\begin{proof} + Let $H$ be a potential $C^2$-function. Then + \begin{equation} + \partial_2 p = \partial_2 \partial_1 H = \partial_1 \partial_2 H = \partial_1 q + \end{equation} +\end{proof} + +\begin{rem} + The above condition is merely necessary! However, for "nice" $D$ it can be considered sufficient. +\end{rem} + +\begin{eg} + Consider + \begin{align*} + \underbrace{(2x + y^2)}_p + \underbrace{2xyy'}_q = 0 && y(1) = 1 + \end{align*} + Then + \begin{align*} + \partial_2 p = 2y && \partial_1 q = 2y + \end{align*} + So $\partial_2 p = \partial_1 q$. If $H$ is a potential function, then + \begin{align*} + \partial_1 H(x, y) &= p(x, y) = 2x + y^2 \\ + \implies H(x, y) &= \int p(x, y) \dd{x} = x^2 + y^2x + G(y) + \end{align*} + and + \begin{align*} + \partial_2 H(x, y) &= q(x, y) = 2xy = 2xy + G'(y) \\ + \implies G(y) &= c + \end{align*} + So the potential function is + \[ + H(x, y) = x^2 + y^2 x + \] + We can insert the initial condition + \[ + H(1, 1) = 2 + \] + So the solution has to fulfil + \[ + x^2 + y(x)^2 x = 2 ~~\forall x \in I + \] + and thus + \[ + y(x) = \pm \sqrt{\frac{2}{x} - x} + \] + Only the positive sign fulfils the initial conditions, so the solution is + \[ + y(x) = \sqrt{\frac{2}{x} - x} + \] + This function is defined on $(-\infty, -\sqrt{2}] \cup (0, \sqrt{2}]$, however due to the initial conditions $(0, \sqrt{2}]$ is the only useful domain. +\end{eg} + +\begin{rem} + If + \[ + p(x, y) + q(x, y)y' = 0 + \] + is not exact one can try and find an "integrating factor", i.e. $h: D \rightarrow \realn$ such that + \[ + h(x, y)p(x, y) + h(x, y)q(x, y)y' = 0 + \] + is exact. A necessary condition is + \[ + \left(\partial_2 h(x, y)\right)p(x, y) + h(x, y) \partial_2 p(x, y) = \left(\partial_1 h(x, y)\right) q(x, y) + h(x, y) \partial_1 q(x, y) + \] + This is a partial differential equation and won't be discussed further in this chapter. +\end{rem} + +\begin{defi}[Ordinary Differential Equation System] + An ordinary differential equation system (ODES) is an equation of the form + \[ + F(x, y, y', \cdots, y^{(n)}) = 0 + \] + with + \[ + F: \realn \times \realn^L \times \realn^L \times \cdots \times \realn^L \longrightarrow \realn^m + \] +\end{defi} + +\begin{eg} + \begin{enumerate}[(i)] + \item Let $z = (z_1, z_2, z_3)$, then + \[ + z'' = -\frac{z}{\norm{z}^3} = -\rec{\norm{z}^2} \frac{z}{\norm{z}} + \] + is the Kepler problem. + + \item The equation + \begin{align*} + b' &= \alpha_1 b - \gamma_1 br \\ + r' &= -\alpha_2 r + \gamma_2 br + \end{align*} + is called the "Lotka-Volterra-Equation" and it models the population of prey and predators. + \end{enumerate} +\end{eg} + +\begin{rem} + The ODES + \[ + F(x, y, y', y'', \cdots, y^{((n)}) = 0 + \] + is equivalent to the ODES of first order + \begin{align*} + F(x, y, y_1, y_2, \cdots, y_{n-1}) = 0 && \begin{cases} + y_1 = y' \\ + y_2 = y_1' \\ + \vdots + \end{cases} + \end{align*} +\end{rem} \end{document} \ No newline at end of file diff --git a/script.pdf b/script.pdf index 14619d566c281b677c1319773712b080ed131ae8..cd6594925d0b61c14dfb635e106d986aa0f9739a 100644 GIT binary patch delta 91418 zcmZsi1ymf%w)dIA2X}XO_uw9!;O_2jVIa6WgGSooi|9f}!x#Zk~R8S}&6c`E}3KI$&3Kt3=iV%v}1hQZQ@Tc_cV*~j( zA$=M!j9>-#H5ttApW5Ui!D%LM&W4<5G)J5)#?rpZ7f+b(5tCyks}@3KLOC+@hzMu& znV0V{rFTV4NK^Kis;G!|KetEle^z$6-2IjvlX6%Fq=TUfL6K*z!IXeim!VT2& z#T}{rBm*oaMeBbe6kkj>Sn$9sed)CwCa|gcPI&9oxH4+vr@nei|jV>>&Q=$tt zv(?Tu>2XiKM&hJROz&qAs?FVAf8Miw$F}6Ct9@#^(iVr0gndvzG9xbov#TQ@fX0!w zZPK}Ph=R*c8%H&&+jAsBD@{52RSG%N5*zud#4uYmBvm6yLa_t3Sm?SHkIq>IAplV^ zmmQ2Q6Zw{UPL4iz+*u{?355>n_FVG=hBc?}7E6~>FJ6Y8XZpz9h{~iPk=1FKT3iM( 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