14.2 FIRST-DEGREE FIRST-ORDER EQUATIONS
Solution method.Check that the equation is an exact differential using (14.7) then
solve using (14.8). Find the functionF(y)by differentiating (14.8) with respect to
yand using (14.6).
14.2.3 Inexact equations: integrating factorsEquations that may be written in the form
A(x, y)dx+B(x, y)dy= 0 but for which∂A
∂y=∂B
∂x(14.9)are known as inexact equations. However, the differentialAdx+Bdycan always
be made exact by multiplying by anintegrating factorμ(x, y), which obeys
∂(μA)
∂y=∂(μB)
∂x. (14.10)
For an integrating factor that is a function of bothxandy,i.e.μ=μ(x, y), there
exists no general method for finding it; in such cases it may sometimes be found
by inspection. If, however, an integrating factor exists that is a function of either
xoryalone then (14.10) can be solved to find it. For example, if we assume
that the integrating factor is a function ofxalone, i.e.μ=μ(x), then (14.10)
reads
μ∂A
∂y=μ∂B
∂x+Bdμ
dx.Rearranging this expression we find
dμ
μ=1
B(
∂A
∂y−∂B
∂x)
dx=f(x)dx,where we requiref(x) also to be a function ofxonly; indeed this provides a
general method of determining whether the integrating factorμis a function of
xalone. This integrating factor is then given by
μ(x)=exp{∫
f(x)dx}
where f(x)=1
B(
∂A
∂y−∂B
∂x). (14.11)
Similarly, ifμ=μ(y)then
μ(y)=exp{∫
g(y)dy}
where g(y)=1
A(
∂B
∂x−∂A
∂y). (14.12)