The order of the differential equation of all parabolas,whose latus rectum is $4a$ and axis is parallel to the $x$-axis,is

The curve $y=(\cos x+y)^{1 / 2}$ satisfies the differential equation

Let the tangent to a curve at point $(x, y)$ intersect the $Y$-axis at point $P$. $A$ line drawn through point $P$ is perpendicular to this tangent and passes through the point $(1, 0)$. The differential equation of the curve is:

If $y = c{e^{{{\sin }^{ - 1}}x}}$,then the corresponding differential equation is:

Verify that the given function $x+y=\tan ^{-1} y$ is a solution of the differential equation $y^{2} y^{\prime}+y^{2}+1=0$.

The differential equation corresponding to the family of curves given by $a x^2+b y^2=1$, where $a$ and $b$ are arbitrary constants, is: