Let us address each part of the question step by step.

Part (a)

The equation given is:

$y^2 + \left(x - \frac{1}{y}\right) \frac{dy}{dx} = 0$

Step 1: Transform to a linear differential equation

Divide through by $y^2$ to simplify:

$1 + \left(x - \frac{1}{y}\right) \frac{1}{y^2} \frac{dy}{dx} = 0$

Let $z = \frac{1}{y}$, so $y = \frac{1}{z}$ and $\frac{dy}{dx} = -\frac{1}{z^2} \frac{dz}{dx}$. Substitute these into the equation:

$1 - (x - z)\frac{1}{z^2}\left(-\frac{dz}{dx}\right) = 0$

Simplify:

$1 + \frac{x - z}{z^2} \frac{dz}{dx} = 0$

Multiply through by $z^2$:

$z^2 + (x - z) \frac{dz}{dx} = 0$

Simplify further:

$z^2 + x \frac{dz}{dx} - z \frac{dz}{dx} = 0$

$z^2 + x \frac{dz}{dx} - z \frac{dz}{dx} = 0$