Question

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

• A. $1$
• B. $2$
• C. $3$
• D. $\frac{3}{2}$

Answer 1

The Correct Option is B

The given differential equation is:

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

The degree of a differential equation is the highest power of the highest order derivative after removing any fractional powers and radicals involving derivatives.

Raise both sides to the power of \(\frac{2}{3}\) to eliminate the fractional exponent:

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

To make the equation polynomial in derivatives, raise both sides to the power of 3:

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

In this form, the highest order derivative is \(\frac{d^2y}{dx^2}\), and its highest power is 2.

Thus, the degree of the differential equation is: 2