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. $\frac{3}{2}$

Hint:

To determine the degree of a differential equation, first remove any radicals or fractional exponents by raising both sides of the equation to the appropriate powers. This ensures the equation is polynomial in terms of its derivatives. Always focus on the highest order derivative and its highest power once the equation is free from any fractional terms. This process helps in clearly identifying the degree.

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

Answer 2

The given differential equation is:

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

Step 1: Understand the concept of the degree of a differential equation:

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

Step 2: Eliminate the fractional exponent:

To eliminate the fractional exponent, raise both sides of the equation to the power of \( \frac{2}{3} \):

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

Step 3: Make the equation polynomial in derivatives:

Now, raise both sides to the power of 3 to eliminate the fractional powers:

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

Step 4: Determine the degree of the differential equation:

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

Conclusion: Thus, the degree of the differential equation is: 2