The given differential equation is $\left( \frac{d^4 y}{d x^4} \right)^2 = \left[ x + \left( \frac{d y}{d x} \right)^2 \right]^3$ $\left( \frac{d^4 y}{d x^4} \right)^2 = x^3 + \left( \frac{d y}{d x} \right)^6 + 3x^2 \left( \frac{d y}{d x} \right)^2 + 3x \left( \frac{d y}{d x} \right)^4$The highest order derivative in the differential equation is $\frac{d^4 y}{d x^4} \Rightarrow$ Order of the given differential equation is 4 .The highest power raised to $\frac{d^4 y}{d x^4}$ is $2 \Rightarrow$ Degree of the given differential equation is 2 .