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Transformation of Homogeneous Equations into Separable Equations

Nonlinear Equations That Can be Transformed Into Separable Equations

We’ve seen that the nonlinear Bernoulli equation can be transformed into a separable equation by the substitution if is suitably chosen. Now let’s discover a sufficient condition for a nonlinear first order differential equation

to be transformable into a separable equation in the same way. Substituting into (eq:2.4.4) yields which is equivalent to If for some function , then (eq:2.4.5) becomes which is separable. After checking for constant solutions such that , we can separate variables to obtain

Homogeneous Nonlinear Equations

In the text we’ll consider only the most widely studied class of equations for which the method of the preceding paragraph works. Other types of equations appear in Exercises exer:2.4.44exer:2.4.51.

The differential equation (eq:2.4.4) is said to be homogeneous if and occur in in such a way that depends only on the ratio ; that is, (eq:2.4.4) can be written as

where is a function of a single variable. For example, and are of the form (eq:2.4.7), with respectively. The general method discussed above can be applied to (eq:2.4.7) with (and therefore . Thus, substituting in (eq:2.4.7) yields and separation of variables (after checking for constant solutions such that ) yields

Before turning to examples, we point out something that you may’ve have already noticed: the definition of homogeneous equation given here isn’t the same as the definition given in Section 2.1, where we said that a linear equation of the form is homogeneous. We make no apology for this inconsistency, since we didn’t create it! Historically, homogeneous has been used in these two inconsistent ways. The one having to do with linear equations is the most important. This is the only section of the book where the meaning defined here will apply.

Since is in general undefined if , we’ll consider solutions of nonhomogeneous equations only on open intervals that do not contain the point .

Substituting into (eq:2.4.8) yields Simplifying and separating variables yields Integrating yields . Therefore and .

Figure figure:2.4.2 shows a direction field and integral curves for (eq:2.4.8).

Figure 1: A direction field and some integral curves for

In the last two examples we were able to solve the given equations explicitly. However, this isn’t always possible, as you’ll see if you attempt the exercises in Trench, Section 2.4.