The degree of the differential equation $\left(1-\left(\frac{d y}{d x}\right)^{2}\right)^{\frac{3}{2}}=k \frac{d^{2} y}{d x^{2}}$ is :

First, let's identify that the highest-order derivative is $\frac{d^2y}{dx^2}$ (second-order).

Next, we need to eliminate the radical by squaring both sides:

$\begin{aligned} & {\left[\left(1-\left(\frac{d y}{d x}\right)^{2}\right)^{\frac{3}{2}}\right]^{2}=\left[k \frac{d^{2} y}{d x^{2}}\right]^{2}} \\ & \left(1-\left(\frac{d y}{d x}\right)^{2}\right)^{3}=k^{2}\left(\frac{d^{2} y}{d x^{2}}\right)^{2}\end{aligned}$

Now the equation is in polynomial form. The highest-order derivative $\frac{d^2y}{dx^2}$ is raised to the power of 2.

Therefore, the degree of this differential equation is 2.

Note: The degree is determined by the highest power of the highest-order derivative after converting to polynomial form. Don't confuse this with the order of the equation (which is also 2 in this case).