The number of arbitrary constants in the general solution of a differential equation of fourth order is:

Determine the order and degree (if defined) of the differential equation $y^{\prime} + y = e^{x}$.

Consider the following differential equations.
$D_1: y=4 \frac{dy}{dx}+3x \frac{dx}{dy}$
$D_2: \frac{d^2y}{dx^2}=\left(3+\left(\frac{dy}{dx}\right)^2\right)^{\frac{4}{3}}$
$D_3: \left[1+\left(\frac{dy}{dx}\right)\right]^2=\left(\frac{dy}{dx}\right)^2$
The ratio of the sum of the orders of $D_1, D_2$ and $D_3$ to the sum of their degrees is

If $a$ and $b$ are respectively the order and degree of the differential equation $y^2(y^{\prime \prime})^2 + 3x(y^{\prime})^{1/3} + x^2y^2 = \sin x$,then:

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

The order and degree of the differential equation $\sqrt[4]{\left(\frac{d^3 y}{d x^3}\right)^5} = \sqrt[3]{\left(\frac{d^2 y}{d x^2}\right)^4}$ are . . . . . . and . . . . . . respectively.