Which of the following transformation reduces the differential equation $\frac{dz}{dx}+\frac{z}{x}log z =\frac{z}{x^2} (log z)^2$ into the form $\frac{du}{dx}+ P (x) u = Q (x)$?

u = (log z)^-1

Dividing the given equation by z (log z)², we get

$\frac{1}{z (log z)^2}\frac{dz}{dx}+\frac{1}{logz}\frac{1}{x}=\frac{1}{x^2}$ … (1)

Writing $\frac{1}{logz}= u$,

we have $\frac{du}{dx}= - (log z)^{-2}\frac{1}{z}\frac{dz}{dx}$

Hence (1) can be written as

$-\frac{du}{dx}+\frac{u}{x}=\frac{1}{x^2} ⇒\frac{du}{dx}-\frac{u}{x}=\frac{-1}{x^2}$

which is the required form with $P (x) =\frac{-1}{x}$ and $Q (x) =\frac{-1}{x^2}$

Hence (C) is the correct answer.