The differential equation found by the elimination of the arbitrary constant $K$ from the equation $y = (x + K)e^{-x}$ is

The differential equation of all circles having center on the $x$-axis and touching the $y$-axis at the origin 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 of radius $a$ is of order:

If the order and degree of the differential equation corresponding to the family of curves $y^2=4a(x+a)$ (where $a$ is a parameter) are $m$ and $n$ respectively,then $m+n^2=$

The differential equation representing the family of parabolas having vertex at the origin and axis along the positive $Y$-axis is