The number of arbitrary constants in the particular solution of a differential equation of fourth order is . . . . . .

The degree of the differential equation $(1 + \frac{dy}{dx})^2 = (\frac{d^3y}{dx^3})^{1/3}$ is . . . . . . .

If $y=a^3 e^{b^2 x+c}$ is the general solution of a differential equation,where $a$ and $c$ are arbitrary constants and $b$ is a fixed constant,then the order of the differential equation is

The differential equation $\left[\frac{1+\left(\frac{dy}{dx}\right)^2}{\left(\frac{d^2y}{dx^2}\right)^{\frac{3}{2}}}\right]^2 = kx$ is of

The differential equation $\frac{d^3y}{dx^3}-5y \frac{dy}{dx}+xy=0$ represents :-

Order of the differential equation of the family of all concentric circles centered at $(h, k)$ is