The differential equation of all lines where the length of the normal from the origin is p and the inclination of the normal is $\alpha$ is... (where p and $\alpha$ are arbitrary constants)

If the solution of the differential equation $(1 + x^3)\dfrac{dy}{dx} + 6x^2y = 1 + x^2$ is $y = \dfrac{1}{(1 + x^3)^s}\left[x + \dfrac{x^p}{p} + \dfrac{x^q}{q} + \dfrac{x^r}{r} + c\right]$, then the LCM of $p, q, r$ and $s$ is...

The solution of the differential equation $\dfrac{dy}{dx} = \cos(x + y)$ is...

The solution of the differential equation $\dfrac{dy}{dx} = \dfrac{x - y}{x + y}$, when $x = 0$ and $y = 0$ represents ....