The differential equation of all straight lines passing through the origin is

The order of the differential equation whose solution is $y=a \cos x+b \sin x+c e^{-x}$ is

$y = ax + b$ is

$f(x, y, c_1, c_2) = 0$ 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 all circles which pass through the origin and whose centres lie on the $y$-axis is:

The differential equation representing the family of curves $x \sin x + y^3 = 4ax$ is