Page 48 - Schaum's Outline of Differential Equations

CHAPTER 5

Exact First-Order

Differential Equations

DEFINING PROPERTIES

A differential equation

is exacl if there exists a function ,i;(.v. y) such that

Test for exactness: If Ml.v, y) and /V(A\ y) are continuous functions and ha\e continuous first partial deriva-
tives on some rectangle of the .vy-plane. then (5./) is exact if and onl\ if

METHOD OF SOLUTION

To solve Eq. (5.1). assuming that it is exact, first solve the equations

for g(x, y). The solution to (5./I is then given implicitly In

where c represents an arbitrary constant.
Equation (5.6) is immediate from Eqs. (J./l and (5.2). If (5.2) is substituted into I.5./I. \\e obtain
dg(x,y(x)) = Q. Integrating this equation (note that we can write 0 as Qtlx), \ve havejdg(x,y(x))=]0dx, which,
in turn, implies (5.6).

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