Question

The degree of differential equation \[ 9 \frac{d^2y}{dx^2} = \left\{ 1 + \left( \frac{dy}{dx} \right)^2 \right\}^{\frac{1}{3}} \text{ is} \]

• A. 1
• B. 6
• C. 3
• D. 2

Hint:

To find the degree of a differential equation, first eliminate all radicals and fractions. Then identify the highest order derivative and note its power. That power is the degree.

Answer 1

The Correct Option is C

We need to find the degree of the given differential equation.

Step 1: Recall the definition of degree of a differential equation.
The degree of a differential equation is defined as:
- The power of the highest order derivative
- After the equation has been made free from radicals and fractions
- Provided the equation is a polynomial in all derivatives

Step 2: Write the given equation.
\[ 9 \frac{d^2y}{dx^2} = \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^{\frac{1}{3}} \]
Step 3: Remove the radical (cube root).
To make the equation free from radicals, we raise both sides to the power 3: \[ \left[ 9 \frac{d^2y}{dx^2} \right]^3 = 1 + \left( \frac{dy}{dx} \right)^2 \]
Step 4: Simplify the left-hand side.
\[ 9^3 \left( \frac{d^2y}{dx^2} \right)^3 = 1 + \left( \frac{dy}{dx} \right)^2 \] \[ 729 \left( \frac{d^2y}{dx^2} \right)^3 = 1 + \left( \frac{dy}{dx} \right)^2 \]
Step 5: Identify the highest order derivative.
The highest order derivative in the equation is \(\frac{d^2y}{dx^2}\) (second order derivative).
In the equation \(729 \left( \frac{d^2y}{dx^2} \right)^3 = 1 + \left( \frac{dy}{dx} \right)^2\), the highest order derivative \(\frac{d^2y}{dx^2}\) appears with power 3.

Step 6: Determine the degree.
The degree is the power of the highest order derivative, which is 3.

Step 7: Conclusion.
The degree of the given differential equation is 3.
Final Answer: (C) 3