The degree of the differential equation

is

The correct option is 2.

Step-by-step Explanation:

To find the order and degree of a differential equation, we must first ensure that the equation is written in a rational form free from any radicals or fractional powers with respect to its derivatives.

The differential equation given in the image is:

$${\left(1-{\left(\frac{dy}{dx}\right)}^2\right)}^{3/2}=k\frac{d^{2}y}{d{x}^2}$$

Notice that the left-hand side of the equation has a fractional exponent of $3/2$. To eliminate this fraction, we square both sides of the equation:

$${\left[{\left(1-{\left(\frac{dy}{dx}\right)}^2\right)}^{3/2}\right]}^2={\left(k\frac{d^{2}y}{d{x}^2}\right)}^2$$

Simplifying this, we get:

$${\left(1-{\left(\frac{dy}{dx}\right)}^2\right)}^3=k^2{\left(\frac{d^{2}y}{d{x}^2}\right)}^2$$

Now, the differential equation is in a polynomial form with respect to its derivatives.
1. Order: The order of a differential equation is the order of the highest-order derivative present in it. Here, the highest derivative is the second derivative, $\frac{d^{2}y}{d{x}^2}$, so the order is 2.
2. Degree: The degree of a differential equation is the power to which the highest-order derivative is raised. In this polynomial form, the highest derivative term $\frac{d^{2}y}{d{x}^2}$ is raised to the power of 2.

Therefore, the degree of the differential equation is 2.