Is this differential equation separable? $$x\frac{dy}{dx}-y^2 = \frac{dy}{dx}+5$$
I have found that this equation is differentiable as shown in the following.
$$x\frac{dy}{dx}-\frac{dy}{dx} = y^2+5$$
$$dy(\frac{x}{dx}-\frac{1}{dx}) = y^2+5$$
$$\frac{x-1}{dx} = \frac{y^2+5}{dy}$$
$$\frac{dx}{x-1} = \frac{dy}{y^2+5}$$
$$dx(\frac{1}{x-1}) = dy(\frac{1}{y^2+5})$$
Is this method valid? Or to be more specific, is the following equation valid?
$$x\frac{dy}{dx}-\frac{dy}{dx} =dy(\frac{x}{dx}-\frac{1}{dx})  $$

It'd be more accurate to write
$$ (x-1)\frac{dy}{dx} = y^2 + 5$$
$$\frac{dy}{dx} = \frac{y^2+5}{x-1}$$
Since the RHS is a product of functions of $x$ and $y$, the equation is separable
