Exact Equations. The equationP(x)y + Q(x)y + R(x)y = 0is said to be exact if it can be

Chapter 3, Problem 41

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Exact Equations. The equationP(x)y + Q(x)y + R(x)y = 0is said to be exact if it can be written in the form[P(x)y] + [f(x)y] = 0,where f(x) is to be determined in terms of P(x), Q(x), and R(x). The latter equation canbe integrated once immediately, resulting in a first order linear equation for y that can besolved as in Section 2.1. By equating the coefficients of the preceding equations and theneliminating f(x), show that a necessary condition for exactness isP(x) Q(x) + R(x) = 0.It can be shown that this is also a sufficient condition.