Consider the differential equation $\frac{d y}{d x}=\frac{1}{a x+4 y+7}$ and the following statements

A. The given differential equation is linear in $x$.

B. The given differential equation is not linear in $y$.

C. The given differential equation is linear in $y$.

D. $e^{a x}$ is the integrating factor of the given differential equation.

Which one of the following options is true?

The solution of the differential equation $x \cos x \frac{d y}{d x}+(x \sin x+\cos x) y=1$ is

If $\alpha$ and $\beta$ are respectively the order and degree of the differential equation for which $a x^2+b y^2=1$ is the general solution, then the eccentricity of the ellipse $\alpha x^2+\beta y^2=1$ is

The solution of the differential equation $x d y-y d x=\sqrt{x^2+y^2} d x$, given that $y=1$ when $x=\sqrt{3}$, is