560 CHAPTER 29. FIRST ORDER SCALAR ODE

and this happens if and only if

xMNx− xMxN = yNMy− yMNy

which happens if and only if

MxNx +MyNy = NyMy +NxMx

if and only ifM (xNx + yNy) = N (yMy + xMx)

But this is true because by Euler’s identity, xNx + yNy = αN and yMy + xMx = αM so theabove is just αNM = αNM. Of course it is assumed that xM+ yN ̸= 0 in the above.

Example 29.6.3 Find the integral curves for(x2 + xy

)dx+

(y2 + x2)dy = 0

Of course this can be written as a homogeneous equation and the technique for solvingthese can be used. However, let’s use this new technique which says that an integratingfactor is

1x(x2 + xy)+ y(y2 + x2)

=1

x3 +2x2y+ y3

Then multiplying by this yields an exact equation.

x2 + xyx3 +2x2y+ y3 dx+

y2 + x2

x3 +2x2y+ y3 dy = 0

Unfortunately, it is too complicated for me to solve this conveniently. However, knowingthat it is exact allows the use of the formula derived in showing that if My = Nx then theequation was exact. Thus the integral curves are of the form∫ x

0M (t,y)dt +N (0,y)

=∫ x

0

t2 + tyt3 +2t2y+ y3 dt +

1y=C

Now we consider an easier one.

Example 29.6.4 Find the integral curves for(xy+ y2)dx+ x2dy = 0

The integrating factor is1

xy(2x+ y)

and so the equation to solve is

1x(2x+ y)

(x+ y)dx+x

y(2x+ y)dy = 0