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