MATH 255 Lecture Notes - Lecture 2: Constant Function, Power Rule, Antiderivative
Document Summary
= f (x; y) (1) is the general form of a rst-order ordinary di erential equation (ode). This is about given a function f (x; y), one is asked to nd a function y = y(x) so that dy(x) dx. If such function y(x) is found, we say that it is a solution of this ode. The general solution for (1) is an expression of all solutions to (1). For example, the general solution to dy dx = y2 is y(x) 0; or y(x) = The initial value problem (ivp) associated with the ode (1) is dy dx = f (x; y) y(x0) = y0 (2) That is, an initial condition y(x0) = y0 (with given x0; y0) is imposed. A solution y(x) of this ivp not only has to satisfy (1), but also has to satisfy this initial condition. If f is good (smooth), then there exists a unique solution to the ivp (2).
