The order of the differential equation whose solution is ${x^2} + {y^2} + 2gx + 2fy + c = 0$ is:

The order and degree of the differential equation ${\left( {1 + 3\frac{{dy}}{{dx}}} \right)^{\frac{2}{3}}} = 4\frac{{{d^3}y}}{{d{x^3}}}$ are

The degree of the differential equation $y(x) = 1 + \frac{dy}{dx} + \frac{1}{2!} \left( \frac{dy}{dx} \right)^2 + \frac{1}{3!} \left( \frac{dy}{dx} \right)^3 + \dots$ is

The order and degree of the differential equation $\left[ 4 + \left( \frac{dy}{dx} \right)^2 \right]^{2/3} = \frac{d^2y}{dx^2}$ are

If $m$ and $n$ are the order and degree of the differential equation $\left( \frac{d^2y}{dx^2} \right)^5 + 4\frac{\left( \frac{d^2y}{dx^2} \right)^3}{\left( \frac{d^3y}{dx^3} \right)} + \frac{d^3y}{dx^3} = x^2 - 1$,then

The order and degree of the differential equation $3 x^2 \frac{d^2 y}{d x^2}-\sin \left(\frac{d^3 y}{d x^3}\right)+\cos (x y)=0$ are