29.6. THE INTEGRATING FACTOR 559

here is something nice which was discovered by Euler back in the 1700s. It is called Euler’sidentity along with the more famous one involving complex numbers.3

Lemma 29.6.2 A function M (x,y) is homogeneous of degree α if M (tx, ty) = tα M (x,y).For such a function,

αM (x,y) = x∂M∂x

(x,y)+ y∂M∂y

(x,y)

Proof: You use the chain rule to differentiate both sides of the equation

M (tx, ty) = tα M (x,y)

with respect to t. Thus

αtα−1M (x,y) = x∂M∂x

(tx, ty)+ y∂M∂y

(tx, ty)

Now let t = 1. ■The reason this is pretty nice is that if you have the equation

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

and both M and N are homogeneous of degree α, then

1xM+ yN

is an integrating factor. Here Mx =∂M∂x . We verify this next. It is so if(

NxM+ yN

)x=

(M

xM+ yN

)y

By the quotient rule, this will be so if and only if

Nx (xM+ yN)−N (M+ xMx + yNx) =

My (xM+ yN)−M (xMy +N + yNy)

In both sides of the above equation, some terms cancel and it follows that the desired resultfollows if and only if

xMNx− (NM+ xMxN) = yNMy− (MN + yMNy)

3Euler (1707-1783) (pronounced “oiler”) was a Swiss mathematician, a student of Johann Bernoulli. He is oneof the most important mathematicians to ever live. He wrote more mathematics than anyone else, some 530 booksand papers in all areas of the subject. His very unusual memory allowed him to continue doing mathematicalresearch even after he went blind in 1766. Many of the ideas in this book are due to him. Like many of the othergreat mathematicians of his time Euler’s interests were not limited to mathematics. His work is also very importantin engineering and physics. A remarkable amount of notation is due to him or popularized by him. Included in thislist is the summation symbol Σ, e,π, i, and f (x). Like many of his time, he was a very religious man who believedthe Bible was inspired. He had incredible insight but like most of us, he made mistakes because he sometimesneglected issues related to convergence. However, the need for this sort of thing was not well understood in histime. Euler died in St. Petersburg.