Question

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

• A. 1
• B. 6
• C. 3
• D. 2

Hint:

Always make sure to clear any radicals or fractional exponents involving derivatives before determining the degree. A common mistake is to state the degree without first converting the equation into a polynomial form of derivatives. The order can be found by simple inspection, but the degree requires this extra step.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The order of a differential equation is the order of the highest derivative appearing in it. The degree of a differential equation is the power of the highest order derivative, provided the equation is a polynomial in its derivatives (i.e., it is free from radicals and fractional powers of the derivatives).
Step 2: Key Approach:
To find the degree, we must first eliminate any fractional powers on the derivative terms. This is typically done by raising both sides of the equation to an appropriate integer power.
Step 3: Detailed Calculation:
The given differential equation is: \[ 9 \frac{d^2y}{dx^2} = \left\{1 + \left(\frac{dy}{dx}\right)^2\right\}^{\frac{3}{2}} \] 1. Identify the highest order derivative:
The derivatives present are \( \frac{dy}{dx} \) (order 1) and \( \frac{d^2y}{dx^2} \) (order 2). The highest order is 2.
2. Eliminate the fractional power:
The equation has a fractional power of \( \frac{3}{2} \). To eliminate it, we need to square both sides of the equation. \[ \left(9 \frac{d^2y}{dx^2}\right)^2 = \left(\left\{1 + \left(\frac{dy}{dx}\right)^2\right\}^{\frac{3}{2}}\right)^2 \] \[ 81 \left(\frac{d^2y}{dx^2}\right)^2 = \left\{1 + \left(\frac{dy}{dx}\right)^2\right\}^3 \] 3. Determine the degree:
Now that the equation is a polynomial in its derivatives, we find the highest power of the highest order derivative. The highest order derivative is \( \frac{d^2y}{dx^2} \), and its power is 2.
Step 4: Final Answer:
The degree of the given differential equation is 2.