Differential Equations and Linear Algebra, 4th Edition

Chapter 1 Summary

In this chapter we have discussed applications of and solution methods for several important types of first-order differential equations, including those that are separable (Section 1.4), linear (Section 1.5), or exact (Section 1.6). In Section 1.6 we also discussed substitution techniques that can sometimes be used to transform a given first-order differential equation into one that is either separable, linear, or exact.

Lest it appear that these methods constitute a “grab bag” of special and unrelated techniques, it is important to note that they are all versions of a single idea. Given a differential equation

$$f(x,y,y\text{′})=0,$$ (1)

we attempt to write it in the form

$$\frac{d}{dx}\left[G(x,y)\right]=0.$$ (2)

It is precisely to obtain the form in ...