The product of the order and degree of the following differential equation :

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

4

$\text{Given differential equation: } x\left(\frac{d^2 y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 + y^2 = 0$

The highest order derivative is $\frac{d^2 y}{dx^2}$, hence the order of the equation is $2$.

The power of the highest order derivative $\frac{d^2 y}{dx^2}$ is $2$, hence the degree of the equation is $2$.

The product of order and degree is $2 \times 2 = 4$.

Final Answer: 4