Let $c_1, c_2, c_3, c_4$ be arbitrary constants. The order of the differential equation,corresponding to $y=c_1 e^x+c_2 e^{\log _{e} x}+c_3 \sin ^2 x-c_4\left(\cos ^2 x-1\right)$ is

$y = ke^{\sin ^{-1} x} + 3$ is a solution of which differential equation?

$f(x, y, c_1, c_2) = 0$ is an equation containing two arbitrary constants $c_1$ and $c_2$. If the differential equation having $f(x, y, c_1, c_2) = 0$ as its general solution is of $k^{\text{th}}$ order,then the differential equation corresponding to $x^k + y^k = c^2$ ($c$ is an arbitrary constant) is

The differential equation of the family of circles passing through the points $(0,2)$ and $(0,-2)$ is

The differential equation representing the family of curves $y^2=2 c(x+\sqrt{c})$,where $c$ is a positive parameter,is of

The order of the differential equation of a family of curves represented by an equation containing four arbitrary constants will be: