29.5. EXACT EQUATIONS 555

First note this is of the form

y′ =(y

x

)2+(y

x

).

Let u = yx so y = xu. Then

u′x+u = u2 +u

and so, separating the variables yields

duu2 =

dxx

Hence−1

u= ln |x|+C

and soyx= u =

1K− ln |x|

where K =−C. Hencey(x) =

xK− ln |x|

PROCEDURE 29.4.2 To solve a homogeneous equation, one which can be placedin the form

y′ = f(y

x

),

do the following:

1. Define a new variable v = y/x. Then y = xv and so y′ = v+ xv′.

2. Plug in to the equation.

v+ xv′ = f (v) , xdvdx

= f (v)− v

dvf (v)− v

=dxx

This is separable. Place∫

before each side and do what it says. Then choose theconstant of integration to satisfy any initial condition which may be present.

29.5 Exact EquationsSometimes you have a differential equation of the form

M (x,y)dx+N (x,y)dy = 0

where Nx = My. In this happy situation, one can find a function of two variables f (x,y)such that

fx (x,y) = M (x,y) , fy (x,y) = N (x,y) (29.3)