\documentclass{article} \input{../macros.tex} \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{Lecture 1} \subsection*{ Welcome to 18.03SC} Here's a list of some MIT courses that list 18.03 Differential Equations as a prerequisite or co-requisite. 2.001 Mechanics and Materials I 2.003 Dynamics and Vibrations 2.005 Thermal-Fluids Engineering 2.016 Hydrodynamics 3.23 Electrical, Optical, and Magnetic Properties of Materials 6.002 Circuits and Electronics 6.021 Quantitative Physiology: Cells and Tissues 8.04 Quantum Physics I 10.301 Fluid Mechanics 12.005 Applications of Continuum Mechanics to Earth, Atmospheric, and Planetary Sciences 16.01 Unified Engineering 18.100 Analysis I 18.330 Introduction to Numerical Analysis 20.309 Biological Engineering II 22.05 Neutron Science and Reactor Physics Altogether 140 courses at MIT list 18.03 as a prerequisite or a co-requisite. \subsection*{What and why} A {\em differential equation} is a relation between a function and its derivatives. Differential equations form the language in which the basic laws of science are expressed. The science tells us how the system at hand changes "from one instant to the next." The challenge addressed by the theory of differential equations is to take this short-term information and obtain information about long-term overall behavior. So the art and practice of differential equations involves the following sequence of steps: one "models" a system (physical, chemical, biological, economic, or even mathematical) by means of a differential equation; one then attempts to gain information about solutions of this equation; and one then translates this mathematical information back into the scientific context. \pagebreak \begin{verbatim} Solve Differential Equation: _______________\ Behavior over time Short term information / /\ \ / \ / Model \ / Interpret \ / \ / \/ Physical World \end{verbatim} A basic example is given by Newton's law, $F = ma$. Here $a =$ acceleration, the second derivative of $x =$ position. Forces don't affect $x$ directly, but only through its derivatives. This is a second order ODE, and we will study second order ODEs extensively later in the course. \subsection*{Separable equations} In this first Unit we will study ODEs involving only the first derivative; first order: $y' = F(x,y)$ . \bigskip {\bf Example 1:} $y' = 2x$. Solution by integrating: $y = x^2 + c$. Notice that there are many solutions. An expression like this, involving a constant ($c$ here) is called the {\em general solution}. The constant is a "constant of integration." \bigskip {\bf Example 2:} $y' = ky$. General Solution: $y = Ce^{kx}$. MEMORIZE THIS -- it's the central example in this course. (In fact, a good definition of the exponential function $e^x$ is that it is the solution of the differential equation $y' = y$ such that $y(0) = 1$ .) \bigskip {\bf Question 1:} What is the general solution to the ODE $dy/dx = 2y+1$ ? \begin{enumerate} \item $y = Ce^{2x} - 1$ \item $y = Ce^{x/2} - 2$ \item $x = y^2 + y + c$ \item $y = e^{x/2} + C$ \item $y = Ce^{2x} - 1$ \item \label{q1ans6} $y = Ce^{2x} - 1/2$ \item $y = e^{2x} + c$ \item None of the above \item Don't know \end{enumerate} The method: ``separation of variables.'' Recall the method: Put all the $x$'s on one side and $y$'s on the other (if possible): $$dy/(2y+1) = dx$$ Integrate both sides: $$(1/2) ln|2y+1| + c1 = x + c2$$ Amalgamate the constants and (if possible) solve for y in terms of x: $$ln|2y+1| = 2x + c , |2y+1| = e^c e^{2x} , 2y+1 = C e^{2x}$$ $$y = C e^{2x} - 1/2$$ So the answer is: \ref{q1ans6}. We can check this!: $ y' = 2 C e^{2x} = 2(y + 1/2) = 2y+1$. It's a nice feature of differential equations in general: it's easy to check your answer! \bigskip {\bf Question 2:} Is $y' + xy = x$ separable? \begin{itemize} \item Yes \item No \item don't know. \end{itemize} Well, $y' = x - xy = x(1-y)$ so $dy/(1-y) = x dx$ : YES. We could go on to solve this, but you can do that on your own. We will see many other methods of solving various types of equations. Unfortunately, most real life equations aren't explicitly solvable, and often you don't actually care as much about the explicit solution as about the general properties. That's one place where today's topic is helpful. \subsection*{Graphical approach} The ODE $y' = F(x,y)$ specifies a derivative -- that is, a slope -- at every point in the plane. This is a {\em direction field} or {\em slope field}. \bigskip {\bf Example:} $y' = 2x$. \begin{center}FIGURE NEEDED\end{center} Notice that the slope $F(x,y)$ does not depend on $y$ here: It is invariant under vertical translation. A {\em solution} of the differential equation is a function whose graph has the given slope at every point it goes through. I drew some. The graphs of solutions are {\em integral curves}. They are vertically nested parabolas. The translation invariance of the direction field is reflected in the fact that a vertical translate of a solution is another solution. To specify a particular solution of an ODE you have to give an {\em initial condition}: when $x$ takes on a certain value, $y$ takes on a specified value. {\bf Example:} $y' = y$. \begin{center}FIGURE NEEDED\end{center} Notice that the slope $F(x,y)$ does not depend on $x$ here: it is invariant under horizontal translation. Graphs of solutions are now horizontal translates of each other. {\bf Example:} $y' = y^2 - x$. This equation does not admit solutions in elementary functions. Nevertheless we can say interesting things about its solutions. To draw the direction field, find where $F(x,y)$ is constant, say $F(x,y) = m$. This is an {\em isocline}. For example, Figure ??? shows that $x = y^2$ is the isocline for $m=0$. \begin{center}FIGURE NEEDED\end{center} When $m = 1$ the isocline is $x = y^2 - 1$. When $m = -1$ the isocline is $x = y^2 + 1$ . The Mathlet Isoclines illustrates more examples. We have seen in action the {\em existence and uniqueness theorem for ODE's}: $y' = F(x,y)$ has exactly one solution such that $y(a) = b$ , for any $(a,b)$ in the region where $F$ is defined. (The solution $y(x)$ may only exist for $x$ near to $a$ .) (You also have to put some technical conditions on $F$.) The E. and U. theorem says that there is just one integral curve through each point:\\ EVERY POINT LIES ON JUST ONE INTEGRAL CURVE: NO CROSSING ALLOWED.\\ The applet makes it look like many solutions coalesce, but this is just a pixel problem. In reality they are separate, but very close to each other. Many seem to bunch up along the bottom branch of the parabola. Can we explain this? I cleared the solutions and drew in just the isoclines $m = -1$ and $m = 0$. Once a solution gets between these two parabolas, it can never escape. The poor thing can't cross the $m = -1$ parabola because it would have to have slope greater than $-1$ when it does; and it can't cross the $m = 0$ parabola because it would have to have slope less than zero when it does. This is a {\em funnel}. So solutions in that region stay in that region: they are trapped between those two parabolas, which are asymptotic as $x \rightarrow \infty$. All these solutions become very close to the function $- sqrt(x)$ for large $x$ . This is an ideal situation! We know, approximately, but with increasing accuracy, about the long term behavior of these solutions, and the answer doesn't depend on initial conditions (as long as you are in this range). This is "stability." (Of course if the solution doesn't get trapped, it's a different story.) Direction fields let you visualize the qualitative behavior of solutions to differential equations, and this is often what you want to know. But we also want to be able to solve ODEs "analytically," that is, using formulas. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{document}