The differential equation satisfied by the family of curves $y = ax \cos \left( \frac{1}{x} + b \right)$,where $a$ and $b$ are parameters,is

The degree of the differential equation whose solution is $y^2=8a(x+a)$ is

Form the differential equation of the family of parabolas having vertex at origin and axis along the positive $y$-axis.

If $m$ and $n$ are the order and degree of the differential equation of the family of parabolas with focus at the origin and $X$-axis as its axis,then $m n-m+n=$

$f(x, y, c_1, c_2) = 0$ 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

If $\sqrt{y}=\cos ^{-1} x$,then it satisfies the differential equation $(1-x^{2}) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}=c$,where $c$ is equal to