ABSTRACT
The notion of a first-order differential equation being “separable” is a natural generaliztion of the notion of a first-oder differential equation being directly integrable. Moreover, a fairly natural modification of the method for solving directly integrable equations yields the basic approach to solving separable equations. However, it cannot be said that the theory of separable equations is just a trivial extension of the theory of directly integrable equations. Certain issues arise with separable equations that do not arise with directly integrable equations. Some of these issues are pertinent to even more general classes of first-order equations, and will arise in later chapters of this text.
The first part of this chapter is devoted to identifying and solving separable first-order differential equations. The rest of the chapter discusses those issues arising with separable equations. Topics in this part include the use of “explicit solutions” versus “implicite solutions”, the related distinctions between “solution curves” and “integral curves”, the possible dependence of the interval of the solution on the initial data, and the possibility of obtaining “false solutions”.
