Question

The degree of differential equation \[ \frac{d^2y}{dx^2} = \left( y + \frac{dy}{dx} \right)^{\frac{1}{5}} \] will be

• A. 2
• B. 5
• C. 1
• D. \( \frac{1}{5} \)

Hint:

To determine the degree of a differential equation, first make sure the equation is polynomial in the derivatives. If there are fractional or irrational powers, eliminate them by raising both sides to the appropriate power.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The degree of a differential equation is defined as the power of the highest-order derivative, provided the equation is polynomial in the derivatives. In cases where there are fractional powers or irrational terms, we first remove such powers to make the equation polynomial.

Step 2: Detailed Explanation:
We start with the given equation: \[ \frac{d^2y}{dx^2} = \left( y + \frac{dy}{dx} \right)^{\frac{1}{5}} \] To eliminate the fractional power, raise both sides of the equation to the power of 5: \[ \left( \frac{d^2y}{dx^2} \right)^5 = y + \frac{dy}{dx} \] Now, the equation is polynomial in the derivatives. The highest-order derivative here is \( \frac{d^2y}{dx^2} \), which is raised to the power of 5. Therefore, the degree of the equation is 5.
Step 3: Final Answer:
The degree of the given differential equation is 5.