The differential equation of the family of curves represented by the equation ${x^2} + {y^2} = {a^2}$ is

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

$A$ family of curves whose equation is the general solution of a differential equation having order $1$ and degree $3$ is ($g, a, c$ are arbitrary constants):

If $c$ and $d$ are arbitrary constants,then $y=e^{2 x}(\cosh \sqrt{2} x+d \sinh \sqrt{2} x)$ is the general solution of the differential equation

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

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