Question

The difference of the order and degree of the differential equation \[ \left(\frac{d^2y}{dx^2} \right)^{-7/2} - \left(\frac{d^3y}{dx^3} \right)^2 - \left(\frac{d^2y}{dx^2} \right)^{-5/2} - \left(\frac{d^4y}{dx^4} \right) = 0 \]

• A. \( 5 \)
• B. \( 3 \)
• C. \( 4 \)
• D. \( 2 \)

Hint:

The order of a differential equation is the highest derivative present. The degree is the power of the highest derivative after making the equation polynomial in derivatives.

Answer 1

The Correct Option is D

Step 1: Identifying Order The order of a differential equation is the highest derivative present. In the given equation, the highest order derivative is \( \frac{d^4y}{dx^4} \), so the order is \( 4 \). 

 

Step 2: Identifying Degree The degree of a differential equation is the power of the highest-order derivative after making the equation polynomial in derivatives. Since the equation contains negative and fractional powers of derivatives, we first remove these. \[ \left(\frac{d^2y}{dx^2} \right)^{-7/2}, \quad \left(\frac{d^2y}{dx^2} \right)^{-5/2} \] These are non-polynomial terms, so we rewrite them appropriately and determine that the degree of the equation is \( 2 \). 

Step 3: Compute the Difference \[ \text{Difference} = \text{Order} - \text{Degree} = 4 - 2 = 2 \]