Introduction of differential equation.

MIT OpenCourseWarehttp://ocw.mit.edu 18.034 Honors Differential Equations Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. LECTURE 0. TERMINOLOGY AND IMPLICIT SOLUTIONS A differential equation (DE) is an equation between specified derivatives of an unknown function, its value, and known quantities and functions. Many physical laws are formulated as differential equations. Ordinary differential equations are differential equations whose unknowns are functions of a single variable. They commonly arise in dynamical systems and electrical engineering. Partial differential equations are differential equations whose unknown depend two or more independent variables. In this course, we focus only on ordinary differential equations. The order of a differential equation is the largest integer n, for which an n-th derivative occurs in the equation. NOTATION. We typically use t or x for independent variables and y or u, v for unknowns, except for the plane systems of (parametric curves), for which we use t as the independent variable and x and y for unknowns. We use the prime to denote the differentiation. For instance, when t is the dy d2yindependent variable and y is the unknown, y� means and y�� means . dt dt2 In this note, we use t for the independent variable and y for the unknown. The most general form of a differential equation of order n is F (t,y,y�, ,y(n))=0.··· A differential equation of order n is said of normal form if it takes the form y(n) = f(t,y,y�, ,y(n−1)).··· Differential equations are usually considered on an open interval I = {t : a 0 the solution curve is the circle of radius √c centered at the origin. Solving (0.2) for y, we obtain the (explicit) solution y = ± c − x2 , which corresponds to the upper and the lower semicircles. These functions are defined for −√c � x � √c, but they are solutions of (0.1) only for −√c