Advanced Engineering Mathematics, 7th Edition

2.1 Solution Curves Without a Solution

INTRODUCTION

Some differential equations do not possess any solutions. For example, there is no real function that satisfies (y′)² + 1 = 0. Some differential equations possess solutions that can be found analytically, that is, solutions in explicit or implicit form found by implementing an equation-specific method of solution. These solution methods may involve certain manipulations, such as a substitution, and procedures, such as integration. Some differential equations possess solutions, but the differential equation cannot be solved analytically. In other words, when we say that a solution of a DE exists, we do not mean that there also exists a method of solution that will produce explicit or implicit ...