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

The differential equation for which $ax + by = 1$ is the general solution 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 of the family of parabolas with vertex at $(0,-1)$ and having axis along the $Y$-axis is

$y=c^{2}+\frac{c}{x}$ is the solution of the differential equation

The differential equation of $y=ae^{bx}$ (where $a$ and $b$ are parameters) is: