ch1: classification, direction fields, existence/uniqueness
This commit is contained in:
Worker Agent
@@ -1,34 +1,428 @@
|
|||||||
\section{Introduction}
|
\section{Classification of Differential Equations}
|
||||||
\label{ch:introduction}
|
|
||||||
|
|
||||||
\subsection{Classification of ODEs}
|
|
||||||
\label{sec:ch01_classification}
|
\label{sec:ch01_classification}
|
||||||
|
|
||||||
% Content goes here
|
A \textbf{differential equation} (DE) is a relation between an unknown function and its derivatives. Differential equations are the language of dynamical systems, appearing in physics, engineering, biology, economics, and throughout the applied sciences.
|
||||||
|
|
||||||
\subsection{Direction Fields}
|
\subsection{Order and Degree}
|
||||||
|
|
||||||
|
\begin{definition}[Order]
|
||||||
|
The \textbf{order} of a differential equation is the order of the highest derivative appearing in the equation.
|
||||||
|
\end{definition}
|
||||||
|
|
||||||
|
\begin{definition}[Degree]
|
||||||
|
The \textbf{degree} of a differential equation is the power of the highest-order derivative, provided the equation is a polynomial in its derivatives.
|
||||||
|
\end{definition}
|
||||||
|
|
||||||
|
\begin{workedexample}
|
||||||
|
\textbf{Determine the order and degree of the following equations:}
|
||||||
|
|
||||||
|
\begin{enumerate}
|
||||||
|
\item $\displaystyle \frac{\diff^3 y}{\diff x^3} + 2\frac{\diff y}{\diff x} = x$
|
||||||
|
|
||||||
|
The highest derivative is $y'''$, so this is a \textbf{third-order} equation. The highest derivative appears to the first power, so the \textbf{degree is 1}.
|
||||||
|
|
||||||
|
\item $\displaystyle \left(\frac{\diff^2 y}{\diff x^2}\right)^{\!2} + \frac{\diff y}{\diff x} = 0$
|
||||||
|
|
||||||
|
The highest derivative is $y''$, so this is \textbf{second-order}. The highest derivative is squared, so the \textbf{degree is 2}.
|
||||||
|
|
||||||
|
\item $\displaystyle \frac{\diff y}{\diff x} = e^{y} + \sin x$
|
||||||
|
|
||||||
|
Highest derivative is $y'$, so \textbf{first-order}. It appears to the first power, so \textbf{degree 1}.
|
||||||
|
\end{enumerate}
|
||||||
|
\end{workedexample}
|
||||||
|
|
||||||
|
\subsection{Linearity}
|
||||||
|
|
||||||
|
\begin{definition}[Linearity]
|
||||||
|
An $n$-th order differential equation is \textbf{linear} if the unknown function $y$ and all of its derivatives appear to the first power and are not multiplied together. A linear $n$-th order DE has the general form
|
||||||
|
\[
|
||||||
|
a_n(x)\,y^{(n)} + a_{n-1}(x)\,y^{(n-1)} + \cdots + a_1(x)\,y' + a_0(x)\,y = g(x),
|
||||||
|
\]
|
||||||
|
where $a_0, a_1, \dots, a_n$ and $g$ are functions of $x$ only.
|
||||||
|
\end{definition}
|
||||||
|
|
||||||
|
If any term violates these conditions, the equation is \textbf{nonlinear}.
|
||||||
|
|
||||||
|
\begin{workedexample}
|
||||||
|
\textbf{Classify as linear or nonlinear:}
|
||||||
|
|
||||||
|
\begin{enumerate}
|
||||||
|
\item $y'' + 3y' + 2y = \cos x$
|
||||||
|
|
||||||
|
All derivatives appear to first power; coefficients depend only on $x$. \textbf{Linear.}
|
||||||
|
|
||||||
|
\item $y'' + y\,y' + y = 0$
|
||||||
|
|
||||||
|
The term $y\,y'$ multiplies $y$ by its derivative. \textbf{Nonlinear.}
|
||||||
|
|
||||||
|
\item $\sin(y) + y' = x$
|
||||||
|
|
||||||
|
The function $y$ appears inside a transcendental function. \textbf{Nonlinear.}
|
||||||
|
\end{enumerate}
|
||||||
|
\end{workedexample}
|
||||||
|
|
||||||
|
\subsection{Homogeneity}
|
||||||
|
|
||||||
|
\begin{definition}[Homogeneity (linear DEs)]
|
||||||
|
A linear differential equation is \textbf{homogeneous} if $g(x) = 0$; i.e., every term contains $y$ or a derivative of $y$. It is \textbf{non-homogeneous} (or inhomogeneous) if $g(x) \neq 0$.
|
||||||
|
\end{definition}
|
||||||
|
|
||||||
|
\begin{workedexample}
|
||||||
|
\begin{enumerate}
|
||||||
|
\item $y'' - 4y = 0$ \quad $\longrightarrow$ \textbf{Homogeneous linear, second-order.}
|
||||||
|
\item $y'' - 4y = e^{2x}$ \quad $\longrightarrow$ \textbf{Non-homogeneous linear, second-order.}
|
||||||
|
\item $y' + y^2 = 0$ \quad $\longrightarrow$ \textbf{Nonlinear} (not classified as homogeneous/heterogeneous in the linear sense).
|
||||||
|
\end{enumerate}
|
||||||
|
\end{workedexample}
|
||||||
|
|
||||||
|
\subsection{Classification Summary}
|
||||||
|
|
||||||
|
\begin{keyresult}
|
||||||
|
\textbf{ODE Classification Checklist:}
|
||||||
|
\begin{enumerate}
|
||||||
|
\item \textbf{Order:} What is the highest derivative? (e.g., $y^{(3)}$ $\Rightarrow$ third order)
|
||||||
|
\item \textbf{Degree:} What power is the highest derivative raised to?
|
||||||
|
\item \textbf{Linearity:} Does $y$ and every derivative appear only to the first power, never multiplied together?
|
||||||
|
\item \textbf{Homogeneity (linear only):} Is $g(x) = 0$? If yes, homogeneous.
|
||||||
|
\end{enumerate}
|
||||||
|
\end{keyresult}
|
||||||
|
|
||||||
|
\begin{table}[htbp]
|
||||||
|
\centering
|
||||||
|
\caption{Classification of example differential equations}
|
||||||
|
\label{tab:ch01_classification_table}
|
||||||
|
\begin{tabular}{@{} l l c c c @{}}
|
||||||
|
\toprule
|
||||||
|
\textbf{Equation} & \textbf{Order} & \textbf{Degree} & \textbf{Linear?} & \textbf{Homogeneous?} \\
|
||||||
|
\midrule
|
||||||
|
$y' + 2y = e^x$ & 1 & 1 & Yes & No \\
|
||||||
|
$y'' - 3y' + 2y = 0$ & 2 & 1 & Yes & Yes \\
|
||||||
|
$(y'')^2 + y' = 0$ & 2 & 2 & No & --- \\
|
||||||
|
$y' = xy + y^2$ & 1 & 1 & No & --- \\
|
||||||
|
$y^{(4)} + y'' = 0$ & 4 & 1 & Yes & Yes \\
|
||||||
|
\bottomrule
|
||||||
|
\end{tabular}
|
||||||
|
\end{table}
|
||||||
|
|
||||||
|
The remainder of this handbook develops solution methods organized by these classifications; see \cref{ch:first_order} for first-order techniques and \cref{ch:qualitative} for qualitative methods.
|
||||||
|
|
||||||
|
% =============================================================================
|
||||||
|
% INITIAL AND BOUNDARY VALUE PROBLEMS
|
||||||
|
% =============================================================================
|
||||||
|
|
||||||
|
\section{Initial and Boundary Value Problems}
|
||||||
|
\label{sec:ch01_ivp_bvp}
|
||||||
|
|
||||||
|
A differential equation by itself typically admits a family of solutions. Additional conditions—initial or boundary values—restrict the family to a specific solution relevant to the physical or mathematical problem.
|
||||||
|
|
||||||
|
\subsection{Initial Value Problems (IVP)}
|
||||||
|
|
||||||
|
\begin{definition}[Initial Value Problem]
|
||||||
|
An \textbf{initial value problem} (IVP) consists of a differential equation together with conditions specifying the value of the unknown function and/or its derivatives at a single point $x_0$:
|
||||||
|
\[
|
||||||
|
y^{(n)} = f(x, y, y', \dots, y^{(n-1)}), \qquad
|
||||||
|
y(x_0) = y_0,\; y'(x_0) = y_0',\; \dots,\; y^{(n-1)}(x_0) = y_0^{(n-1)}.
|
||||||
|
\]
|
||||||
|
An $n$-th order IVP requires $n$ initial conditions.
|
||||||
|
\end{definition}
|
||||||
|
|
||||||
|
\begin{workedexample}
|
||||||
|
\textbf{IVP examples:}
|
||||||
|
|
||||||
|
\begin{enumerate}
|
||||||
|
\item First-order IVP:
|
||||||
|
\[
|
||||||
|
\frac{\diff y}{\diff x} = 2x, \qquad y(0) = 3.
|
||||||
|
\]
|
||||||
|
Integrating: $y(x) = x^2 + C$. The condition $y(0) = 3$ gives $C = 3$, so $y(x) = x^2 + 3$.
|
||||||
|
|
||||||
|
\item Second-order IVP:
|
||||||
|
\[
|
||||||
|
\frac{\diff^2 y}{\diff x^2} = 6x, \qquad y(0) = 1,\; y'(0) = -2.
|
||||||
|
\]
|
||||||
|
Integrating twice: $y'(x) = 3x^2 + C_1$, $y(x) = x^3 + C_1 x + C_2$. The conditions give $C_1 = -2$ and $C_2 = 1$, so $y(x) = x^3 - 2x + 1$.
|
||||||
|
\end{enumerate}
|
||||||
|
\end{workedexample}
|
||||||
|
|
||||||
|
IVPs model \textbf{evolution} problems: given the state of a system at an initial time, predict its future behavior. Typical applications include population growth, radioactive decay, projectile motion, and electrical circuits driven by initial charge/charge rate.
|
||||||
|
|
||||||
|
\subsection{Boundary Value Problems (BVP)}
|
||||||
|
|
||||||
|
\begin{definition}[Boundary Value Problem]
|
||||||
|
A \textbf{boundary value problem} (BVP) consists of a differential equation together with conditions at two or more distinct points:
|
||||||
|
\[
|
||||||
|
y^{(n)} = f(x, y, y', \dots, y^{(n-1)}), \qquad
|
||||||
|
y(a) = \alpha,\; y(b) = \beta, \; \dots
|
||||||
|
\]
|
||||||
|
\end{definition}
|
||||||
|
|
||||||
|
\begin{workedexample}
|
||||||
|
\textbf{Second-order BVP:}
|
||||||
|
\[
|
||||||
|
\frac{\diff^2 y}{\diff x^2} = 0, \qquad y(0) = 0,\; y(1) = 1.
|
||||||
|
\]
|
||||||
|
Integrating: $y(x) = C_1 x + C_2$. The conditions give $C_2 = 0$ and $C_1 = 1$, so $y(x) = x$.
|
||||||
|
\end{workedexample}
|
||||||
|
|
||||||
|
BVPs typically arise in \textbf{steady-state} or \textbf{spatial} problems: heat distribution along a rod, deflection of a beam, or electrostatic potential between electrodes. Unlike IVPs, BVPs may have no solution, a unique solution, or infinitely many solutions.
|
||||||
|
|
||||||
|
\subsection{Comparison}
|
||||||
|
|
||||||
|
\begin{table}[htbp]
|
||||||
|
\centering
|
||||||
|
\caption{IVP vs.\ BVP comparison}
|
||||||
|
\label{tab:ch01_ivp_bvp}
|
||||||
|
\begin{tabular}{@{} l l l l @{}}
|
||||||
|
\toprule
|
||||||
|
& \textbf{IVP} & \textbf{BVP} & \\
|
||||||
|
\midrule
|
||||||
|
\textbf{Conditions at:} & Single point $x_0$ & Two or more distinct points \\
|
||||||
|
\textbf{Typical problem:} & Time evolution, dynamics & Steady state, spatial distribution \\
|
||||||
|
\textbf{Existence:} & Usually guaranteed (Picard--Lindel\"of) & Not guaranteed \\
|
||||||
|
\textbf{Uniqueness:} & Usually guaranteed & May fail or have multiple solutions \\
|
||||||
|
\textbf{Example:} & $y' = y,\; y(0) = 1$ & $y'' = 0,\; y(0)=0,\; y(1)=1$ \\
|
||||||
|
\bottomrule
|
||||||
|
\end{tabular}
|
||||||
|
\end{table}
|
||||||
|
|
||||||
|
% =============================================================================
|
||||||
|
% DIRECTION FIELDS
|
||||||
|
% =============================================================================
|
||||||
|
|
||||||
|
\section{Direction Fields}
|
||||||
\label{sec:ch01_direction_fields}
|
\label{sec:ch01_direction_fields}
|
||||||
|
|
||||||
% Content goes here
|
A direction field provides a geometric picture of the solution family of a first-order ODE without solving it analytically.
|
||||||
|
|
||||||
\subsection{Existence and Uniqueness}
|
\subsection{Construction}
|
||||||
|
|
||||||
|
For a first-order ODE $\displaystyle \frac{\diff y}{\diff x} = f(x, y)$, the direction field is constructed as follows:
|
||||||
|
|
||||||
|
\begin{enumerate}
|
||||||
|
\item Choose a rectangular grid of points $(x_i, y_j)$ in the $xy$-plane.
|
||||||
|
\item At each grid point, compute the slope $m_{ij} = f(x_i, y_j)$.
|
||||||
|
\item Draw a short line segment centered at $(x_i, y_j)$ with slope $m_{ij}$.
|
||||||
|
\end{enumerate}
|
||||||
|
|
||||||
|
Solution curves are trajectories that are everywhere tangent to the direction field. Reading the field visually reveals qualitative features: equilibrium solutions (horizontal segments), asymptotic behavior, and regions of rapid growth or decay.
|
||||||
|
|
||||||
|
\begin{hintbox}
|
||||||
|
Look for \textbf{isoslopes}: curves along which $f(x, y)$ is constant. For $\frac{\diff y}{\diff x} = x - y$, isoslopes are the lines $y = x - c$, along which every segment has the same slope $c$.
|
||||||
|
\end{hintbox}
|
||||||
|
|
||||||
|
\subsection{Direction Field Diagram}
|
||||||
|
|
||||||
|
The following TikZ figure shows the direction field for $\displaystyle \frac{\diff y}{\diff x} = x - y$ over the region $[-2, 2] \times [-2, 2]$, together with two representative solution curves.
|
||||||
|
|
||||||
|
\begin{figure}[htbp]
|
||||||
|
\centering
|
||||||
|
\begin{tikzpicture}[scale=1.3]
|
||||||
|
% Axes
|
||||||
|
\draw[->] (-2.3, 0) -- (2.3, 0) node[below right] {$x$};
|
||||||
|
\draw[->] (0, -2.3) -- (0, 2.3) node[above left] {$y$};
|
||||||
|
|
||||||
|
% Direction field: dy/dx = x - y
|
||||||
|
\foreach \x in {-2,-1.6,-1.2,-0.8,-0.4,0,0.4,0.8,1.2,1.6,2} {
|
||||||
|
\foreach \y in {-2,-1.6,-1.2,-0.8,-0.4,0,0.4,0.8,1.2,1.6,2} {
|
||||||
|
\pgfmathsetmacro{\slope}{\x - \y}
|
||||||
|
\pgfmathsetmacro{\dx}{0.15}
|
||||||
|
\pgfmathsetmacro{\dy}{\slope * 0.15}
|
||||||
|
% Clamp to avoid excessively long segments
|
||||||
|
\pgfmathsetmacro{\len}{sqrt(\dx*\dx + \dy*\dy)}
|
||||||
|
\pgfmathsetmacro{\scale}{min(0.35 / max(\len,0.001), 1)}
|
||||||
|
\pgfmathsetmacro{\sdx}{\dx * \scale}
|
||||||
|
\pgfmathsetmacro{\sdy}{\dy * \scale}
|
||||||
|
\draw[gray!70, very thin] (\x-\sdx,\y-\sdy) -- (\x+\sdx,\y+\sdy);
|
||||||
|
}
|
||||||
|
}
|
||||||
|
|
||||||
|
% Solution curve 1: y = x - 1 + 2*e^{-x}, passing through (0,1)
|
||||||
|
\draw[SteelBlue, thick, domain=-2:2, samples=80, smooth]
|
||||||
|
plot (\x, {\x - 1 + 2*exp(-\x)});
|
||||||
|
|
||||||
|
% Solution curve 2: y = x - 1 - 2*e^{-x}, passing through (0,-3)
|
||||||
|
\draw[SteelBlue!60, thick, domain=-1:2, samples=80, smooth]
|
||||||
|
plot (\x, {\x - 1 - 2*exp(-\x)});
|
||||||
|
|
||||||
|
% Equilibrium line y = x (slope = 0 everywhere)
|
||||||
|
\draw[ForestGreen!70, dashed, domain=-2:2, samples=40]
|
||||||
|
plot (\x, {\x});
|
||||||
|
\node[ForestGreen!80, font=\footnotesize] at (1.8, 1.2) {$y=x$};
|
||||||
|
|
||||||
|
% Tick marks
|
||||||
|
\foreach \t in {-2,-1,1,2} {
|
||||||
|
\draw (\t, 0.05) -- (\t, -0.05) node[below, font=\tiny] {$\t$};
|
||||||
|
\draw (0.05, \t) -- (-0.05, \t) node[left, font=\tiny] {$\t$};
|
||||||
|
}
|
||||||
|
\end{tikzpicture}
|
||||||
|
\caption{Direction field for $\frac{\diff y}{\diff x} = x - y$. Blue curves are solution trajectories. The dashed green line $y = x$ is the locus of zero slope (equilibrium for the shifted system).}
|
||||||
|
\label{fig:ch01_direction_field}
|
||||||
|
\end{figure}
|
||||||
|
|
||||||
|
\subsection{Sketching Solution Curves}
|
||||||
|
|
||||||
|
Given a direction field, individual solution curves can be sketched by hand:
|
||||||
|
|
||||||
|
\begin{enumerate}
|
||||||
|
\item \textbf{Trace the flow:} Start at a point and follow the segments, drawing a smooth curve tangent to the local arrows.
|
||||||
|
\item \textbf{Respect the grid:} The curve should never cross a segment at an angle significantly different from the segment's slope.
|
||||||
|
\item \textbf{Identify special features:} Equilibrium points (zero slope), asymptotes, and inflection points often emerge from the field pattern.
|
||||||
|
\end{enumerate}
|
||||||
|
|
||||||
|
\begin{workedexample}
|
||||||
|
\textbf{Sketch the solution of $\frac{\diff y}{\diff x} = x - y$ passing through $(0, 1)$.}
|
||||||
|
|
||||||
|
\begin{enumerate}
|
||||||
|
\item At $(0, 1)$, the slope is $f(0, 1) = 0 - 1 = -1$. Draw a segment with slope $-1$.
|
||||||
|
\item Moving right: at $(1, 0.5)$ (approximately), the slope is $1 - 0.5 = 0.5$. The curve bends upward.
|
||||||
|
\item The line $y = x$ is where the slope is zero. For $y > x$, the slope is negative (curves bend down toward $y = x$). For $y < x$, the slope is positive (curves bend up toward $y = x$).
|
||||||
|
\item Thus the line $y = x$ acts as an attractor. The solution starting at $(0, 1)$ approaches the line $y = x$ as $x \to \infty$.
|
||||||
|
|
||||||
|
\end{enumerate}
|
||||||
|
|
||||||
|
\textbf{Verification:} The analytic solution is $y(x) = x - 1 + 2e^{-x}$. Indeed, $\lim_{x \to \infty} (y(x) - x) = -1$, and the transient $2e^{-x}$ decays, consistent with the field.
|
||||||
|
\end{workedexample}
|
||||||
|
|
||||||
|
Direction fields also underpin numerical methods such as Euler's method (see \cref{ch:qualitative}).
|
||||||
|
|
||||||
|
% =============================================================================
|
||||||
|
% EXISTENCE AND UNIQUENESS
|
||||||
|
% =============================================================================
|
||||||
|
|
||||||
|
\section{Existence and Uniqueness}
|
||||||
\label{sec:ch01_existence_uniqueness}
|
\label{sec:ch01_existence_uniqueness}
|
||||||
|
|
||||||
% Content goes here
|
A fundamental question before attempting to solve any differential equation is: \emph{does a solution exist?} and \emph{is it unique?} The Picard--Lindel\"of theorem provides the foundational answer for first-order IVPs.
|
||||||
|
|
||||||
\subsection{Summary}
|
\subsection{The Picard--Lindel\"of Theorem}
|
||||||
|
|
||||||
|
\begin{theorem}[Picard--Lindel\"of]
|
||||||
|
\label{thm:picard_lindelof}
|
||||||
|
Consider the first-order IVP
|
||||||
|
\[
|
||||||
|
\frac{\diff y}{\diff x} = f(x, y), \qquad y(x_0) = y_0.
|
||||||
|
\]
|
||||||
|
Let $R = \{(x, y) : |x - x_0| \leq a,\; |y - y_0| \leq b\}$ be a closed rectangle centered at $(x_0, y_0)$ with $a, b > 0$.
|
||||||
|
|
||||||
|
If the following two conditions hold:
|
||||||
|
\begin{enumerate}
|
||||||
|
\item \textbf{Continuity:} $f(x, y)$ is continuous on $R$;
|
||||||
|
\item \textbf{Lipschitz condition:} There exists a constant $L > 0$ such that for all $(x, y_1), (x, y_2) \in R$,
|
||||||
|
\[
|
||||||
|
|f(x, y_1) - f(x, y_2)| \leq L\,|y_1 - y_2|.
|
||||||
|
\]
|
||||||
|
\end{enumerate}
|
||||||
|
Then there exists an interval $|x - x_0| \leq h > 0$ (with $h \leq a$) on which a \textbf{unique} solution $y = \varphi(x)$ exists.
|
||||||
|
\end{theorem}
|
||||||
|
|
||||||
|
\begin{remark}
|
||||||
|
A \textbf{sufficient} (but not necessary) condition for the Lipschitz condition is that $\pd{f}{y}$ exists and is bounded on $R$. In that case, one may take $L = \sup\limits_{(x, y) \in R} \bigl|\pd{f}{y}\bigr|$.
|
||||||
|
\end{remark}
|
||||||
|
|
||||||
|
\subsection{The Lipschitz Condition}
|
||||||
|
|
||||||
|
The Lipschitz condition essentially requires that $f$ does not change too rapidly with respect to $y$. Geometrically, it prevents solution curves from diverging too quickly from one another.
|
||||||
|
|
||||||
|
\begin{workedexample}
|
||||||
|
\textbf{Verify the Lipschitz condition for $f(x, y) = 2xy$ on the rectangle $R = [-1, 1] \times [-1, 1]$.}
|
||||||
|
|
||||||
|
Compute the partial derivative:
|
||||||
|
\[
|
||||||
|
\pd{f}{y} = 2x.
|
||||||
|
\]
|
||||||
|
On $R$, $|x| \leq 1$, so $\bigl|\pd{f}{y}\bigr| \leq 2$. Thus $f$ satisfies a Lipschitz condition with $L = 2$ on $R$. By \cref{thm:picard_lindelof}, the IVP $y' = 2xy,\; y(0) = 1$ has a unique solution on some interval around $x = 0$.
|
||||||
|
\end{workedexample}
|
||||||
|
|
||||||
|
\begin{workedexample}
|
||||||
|
\textbf{Non-uniqueness when the Lipschitz condition fails.}
|
||||||
|
|
||||||
|
Consider the IVP
|
||||||
|
\[
|
||||||
|
\frac{\diff y}{\diff x} = y^{2/3}, \qquad y(0) = 0.
|
||||||
|
\]
|
||||||
|
Here $f(y) = y^{2/3}$. The partial derivative with respect to $y$ is
|
||||||
|
\[
|
||||||
|
\pd{f}{y} = \tfrac{2}{3}\,y^{-1/3},
|
||||||
|
\]
|
||||||
|
which is \textbf{unbounded} as $y \to 0$. The Lipschitz condition fails in any rectangle containing $y = 0$.
|
||||||
|
|
||||||
|
Indeed, there are infinitely many solutions:
|
||||||
|
\[
|
||||||
|
y(x) = 0 \quad \text{(trivial solution)},
|
||||||
|
\]
|
||||||
|
and for any $c \geq 0$,
|
||||||
|
\[
|
||||||
|
y(x) =
|
||||||
|
\begin{cases}
|
||||||
|
0, & 0 \leq x \leq c, \\
|
||||||
|
\dfrac{(x - c)^3}{27}, & x > c,
|
||||||
|
\end{cases}
|
||||||
|
\]
|
||||||
|
all satisfy the IVP. The solution branches off from the trivial solution at any chosen point $c$.
|
||||||
|
\end{workedexample}
|
||||||
|
|
||||||
|
\begin{hintbox}
|
||||||
|
A quick test for potential non-uniqueness: check whether $\pd{f}{y}$ blows up at or near the initial condition point. If it does, the Lipschitz condition likely fails.
|
||||||
|
\end{hintbox}
|
||||||
|
|
||||||
|
\subsection{Interval of Validity}
|
||||||
|
|
||||||
|
\begin{definition}[Interval of Validity]
|
||||||
|
The \textbf{interval of validity} (or interval of existence) of a solution $y = \varphi(x)$ to an IVP is the \textbf{largest} open interval $I$ containing $x_0$ on which the solution exists and is differentiable.
|
||||||
|
\end{definition}
|
||||||
|
|
||||||
|
The interval of validity may be restricted by:
|
||||||
|
\begin{itemize}
|
||||||
|
\item Singularities in the coefficients of the equation;
|
||||||
|
\item Points where the Lipschitz condition breaks down;
|
||||||
|
\item Points where the solution itself becomes unbounded (blow-up).
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
\begin{workedexample}
|
||||||
|
\textbf{Find the interval of validity for $y' = y^2,\; y(0) = 1$.}
|
||||||
|
|
||||||
|
This is separable: $\displaystyle \frac{\diff y}{y^2} = \diff x \;\Rightarrow\; -\frac{1}{y} = x + C$. With $y(0) = 1$, we get $C = -1$, so $y(x) = \frac{1}{1 - x}$.
|
||||||
|
|
||||||
|
The solution has a vertical asymptote at $x = 1$. Since the initial point is $x_0 = 0$, the interval of validity is
|
||||||
|
\[
|
||||||
|
(-\infty, 1).
|
||||||
|
\]
|
||||||
|
The solution blows up at $x = 1$, so it cannot be extended beyond this point.
|
||||||
|
\end{workedexample}
|
||||||
|
|
||||||
|
\begin{workedexample}
|
||||||
|
\textbf{Interval of validity for $y' = \sqrt{y},\; y(0) = 1$.}
|
||||||
|
|
||||||
|
Separating: $\displaystyle \frac{\diff y}{\sqrt{y}} = \diff x \;\Rightarrow\; 2\sqrt{y} = x + C$. With $y(0) = 1$, we get $C = 2$, so $y(x) = \left(\frac{x + 2}{2}\right)^2 = \frac{(x+2)^2}{4}$.
|
||||||
|
|
||||||
|
This solution is defined and differentiable for all $x \in \R$, so the interval of validity is $(-\infty, \infty)$. Note that although the Lipschitz condition fails at $y = 0$ (since $\pd{f}{y} = \frac{1}{2\sqrt{y}}$ is unbounded there), the particular solution passing through $y(0) = 1$ never reaches $y = 0$ for $x > -2$, so uniqueness is preserved along this trajectory.
|
||||||
|
\end{workedexample}
|
||||||
|
|
||||||
|
\section{Summary}
|
||||||
\label{sec:ch01_summary}
|
\label{sec:ch01_summary}
|
||||||
|
|
||||||
% Summary table at end of chapter
|
|
||||||
\begin{table}[htbp]
|
\begin{table}[htbp]
|
||||||
\centering
|
\centering
|
||||||
\caption{Chapter Summary}
|
\caption{Chapter 1 summary: key concepts in differential equations}
|
||||||
\label{tab:ch01_summary}
|
\label{tab:ch01_summary}
|
||||||
\begin{tabular}{l l}
|
\begin{tabular}{@{} p{3.8cm} p{5.5cm} @{}}
|
||||||
\toprule
|
\toprule
|
||||||
\textbf{Concept} & \textbf{Key Formula/Method} \\
|
\textbf{Concept} & \textbf{Key Definition / Formula} \\
|
||||||
\midrule
|
\midrule
|
||||||
TBD & TBD \\
|
\textbf{Order} & Highest derivative present (e.g., $y'''$ $\Rightarrow$ 3rd order) \\
|
||||||
\bottomrule
|
\textbf{Degree} & Power of the highest-order derivative when polynomial in derivatives \\
|
||||||
\end{tabular}
|
\textbf{Linear DE} & $a_n(x)y^{(n)} + \dots + a_1(x)y' + a_0(x)y = g(x)$ \\
|
||||||
|
\textbf{Nonlinear DE} & $y$ or derivatives appear to non-integer powers, multiplied together, or inside nonlinear functions \\
|
||||||
|
\textbf{Homogeneous (linear)} & $g(x) = 0$ in the linear form \\
|
||||||
|
\textbf{IVP} & DE + conditions at a single point: $y(x_0) = y_0, \dots$ \\
|
||||||
|
\textbf{BVP} & DE + conditions at two or more distinct points \\
|
||||||
|
\textbf{Direction field} & Plot of short line segments with slope $f(x_i, y_j)$ at grid points \\
|
||||||
|
\textbf{Picard--Lindel\"of} & Continuity + Lipschitz $\Rightarrow$ local existence and uniqueness \\
|
||||||
|
\textbf{Lipschitz condition} & $|f(x, y_1) - f(x, y_2)| \leq L|y_1 - y_2|$; sufficient: $\bigl|\pd{f}{y}\bigr| \leq L$ \\
|
||||||
|
\textbf{Interval of validity} & Largest interval containing $x_0$ where the solution exists and is unique \\
|
||||||
|
\textbf{Non-uniqueness example} & $y' = y^{2/3},\; y(0) = 0$ admits infinitely many solutions \\
|
||||||
|
\bottomrule
|
||||||
|
\end{tabular}
|
||||||
\end{table}
|
\end{table}
|
||||||
|
|
||||||
|
The chapters that follow build directly on these foundations. Solution methods for first-order equations are developed in \cref{ch:first_order}; qualitative analysis and numerical methods appear in \cref{ch:qualitative}; and linear second-order theory begins in \cref{ch:second_order_homogeneous}.
|
||||||
|
|||||||
Reference in New Issue
Block a user