18.03 Differential Equations

In this article I show how much time I spend at each session of this course, this is automatic system marked with timestamp with all hours This timetables includes understanding the concepts and creating anki cards ...

In Unit I, we will study ordinary differential equations (ODE’s) involving only the first derivative. \(y’ = F(x, y)\) The first session covers some of the conventions and prerequisites for the course. After that we will focus on first order differential equations. We’ll start by defining differential equations and seeing a few well known ones from science and engineering. Very quickly we will learn about the three main ways of approaching ODE’s: Analytic (in symbols), geometric (with pictures and graphs), and numerical (with the computer). A large part of this unit (and this course) will be spent on linear ODE’s. These are fundamental to the understanding of all ODE’s and among the most tractable. Last Modification: 2020-09-16 Wed 19:55 ...

In this unit we learn how to solve constant coefficient second order linear differential equations, and also how to interpret these solutions when the DE is modeling a physical system. The language and ideas we introduced for first order linear constant coefficient DE’s carry forward to the second order case—in particular, the breakdown into the homogeneous and inhomogeneous cases; the input-response language for the inhomogeneous case; and the general form of solutions as \(x = x_h + x_p\) where \(x_h\) is the general solution to the homogeneous equation, and xp is any particular solution to the inhomogeneous equation. We first learn how to solve the homogeneous equation \(ax’’ + bx’ + cx = 0\) As in the first order case, the solutions will be exponential functions. In the second order case, however, the exponential functions can be either real or complex, so that we need to use the complex arithmetic and complex exponentials we developed in the last unit. For the second order inhomogeneous DE \(ax’’ + bx’ + cx = f(t)\) we will concentrate much of our attention on the important case of a sinusoidal signal \(f(t) = B \cos(\omega t)\), and again see how we can simplify the calculations and achieve a better understanding of the solution methods and the solutions themselves by using complex exponentials and arithmetic. We will run an interpretation of the results as the behavior of a physical system in parallel with the mathematical methods and formulas, in order to have a concrete way to understand and visualize what these results mean. We will use a spring-mass mechanical system for the most part, since it is easy to picture. At the end of the unit we will also explore the application of this theory to electrical circuits, which is another very important use of these mathematical methods. Last Modification: 2020-09-16 Wed 21:03 ...

In this unit we will learn two new ways to represent certain types of functions, and these will help us solve linear time invariant (LTI) DE’s with these functions as inputs. We start with Fourier series, which are a way to write periodic functions as sums of sinusoids. In Unit Two we learned how to solve a constant coefficient linear ODE with sinusoidal input. Now using Fourier series and the superposition principle we will be able to solve these equations with any periodic input. Next we will study the Laplace transform. This operation transforms a given function to a new function in a different independent variable. For example, the Laplace transform of \(f(t) = \cos(3t)\) is \(F(s) = s/ (s^2 + 9)\). If we think of $f(t)$$ as an input signal, then the key fact is that its Laplace transform \(F(s)\) represents the same signal viewed in a different way. The Laplace transform converts a DE for the function \(x(t)\) into an algebraic equation for its Laplace transform \(X(s)\). Then, once we solve for \(X(s)\) we can recover \(x(t)\). In the course of this unit, two important ideas will be introduced. The first is the convolution product of two functions. At first meeting this operation may seem a bit strange. Nonetheless, as we will see, it arises naturally, and the Laplace transform will allow us to work easily with it. The second important idea is the delta function. Up to now all inputs to our systems have caused small changes in a small amount of time. An impulse is an input that causes a sudden jump in the system. For example, a sharp blow to a mass will cause its momentum to jump. The delta function is a mathematical idealization of an impulse and one which allows us to handle DE’s with these types of inputs. Last Modification: 2020-09-16 Wed 22:56 ...

In this unit we study systems of differential equations. A system of ODE’s means a DE with one independent variable but more than one dependent variable, for example: \({\displaystyle x’ = x + y}\), \({\displaystyle y’ = x^2 - y - t}\) is a 2x2 system of DE’s for the two functions \(x = x(t)\) and \(y = y(t)\). As usual, we start with the linear case. Even for linear systems, though, it turns out that efficient solution methods require some new techniques, namely the machinery of matrix-vector algebra. A small investment in this background material yields an excellent return, giving both the linear theory in the general case and also the explicit computational methods for the solutions in the constant-coefficient case. We finish this unit by showing some of the qualitative theory of DE’s for systems, linear and non-linear. Qualitative theory means finding out information about the solutions directly from the DE without actually having to solve it. We start with the linear case, and then show how we can use the results for linear constant-coefficient systems to gain information about certain non-linear systems using a technique called linearization. ...