The differential equation of the family of circles whose center lies on the $x$-axis is:

If the order of a differential equation $\frac{d^2 y}{d x^2}-2\left(\frac{d y}{d x}\right)^3+\sin \left(\frac{d y}{d x}\right)+y=0$ is $l$ and the degree of the differential equation $\left(1+\frac{d^2 y}{d x^2}\right)^{\frac{2}{3}}=\left[2-\left(\frac{d y}{d x}\right)^3\right]^{\frac{3}{2}}$ is $m$, then the differential equation corresponding to the family of curves $y=A x^l+B e^{m x}$, where $A$ and $B$ are arbitrary constants, is

The differential equation representing the family of ellipses having foci either on the $x$-axis or on the $y$-axis,centered at the origin,and passing through the point $(0,3)$ is:

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 $m+n^2=$

Form the differential equation of the family of lines $y = mx + \frac{4}{m}$ by eliminating the arbitrary constant $m$.

The differential equation of $3y = \sqrt{x + c}$ is