Ordinary Differential Equations

Abstract

The essential ideas of the method occur for first-order equations and these are discussed first. For first-order equations of first degree, which form the main subject matter of the first part of this book, the difference between the case when a variable is missing in the right hand side and the general case should be noted: (1.0-1) $$ frac{{dy}} {{dt}} = F(x,y)\quad general, $$ (1.0-2) $$ frac{{dy}} {{dx}} = F(x)\quad y\,mis\sin ng\ $$ . In the general case the complete integration is represented by all the integral curves in the (x,y) plane, one curve passing through each nonsingular point (Fig. 1.0–1) according to the local direction field at each point P; these form ∞1 (number of) curves.(1) Their construction demands in general ∞1 integrations.