The sum of order and degree of differential equation $2\sqrt{\frac{d^2y}{dx^2}}+(\frac{dy}{dx})^3=0$ is:

3

Given differential equation

$2\sqrt{\frac{d^2y}{dx^2}}+\left(\frac{dy}{dx}\right)^3=0$

Remove radical to define degree

$2\sqrt{\frac{d^2y}{dx^2}}=-\left(\frac{dy}{dx}\right)^3$

Square both sides

$4\frac{d^2y}{dx^2}=\left(\frac{dy}{dx}\right)^6$

Highest order derivative is $\frac{d^2y}{dx^2}$

Order $=2$

Power of highest order derivative is $1$

Degree $=1$

Sum of order and degree

$=2+1=3$

The required sum is $3$.