Explanation

The equation of the type $y = x frac{dy}{dx} + f frac{dy}{dx}$  can be identified by comparing it to the differential equation form:

$y = x frac{d^2y}{dx^2} + f frac{dy}{dx}$

In this form, the equation involves both the second derivative $frac{d^2y}{dx^2}$ and the first derivative $frac{dy}{dx}$ with coefficients $x$ and $f$ respectively.

Therefore, among the options provided, the equation that fits this type is:

$y = x frac{d^2y}{dx^2} + f frac{dy}{dx}$

This equation indicates a relationship where $y$ is expressed as a combination involving both $frac{d^2y}{dx^2}$ and $frac{dy}{dx}$, with coefficients $x$ and $f$ respectively.