Question

The order and degree of the differential equation \[ \left[ \left( \frac{d^2 y}{dx^2} \right)^2 - 1 \right]^2 = \frac{dy}{dx} \text{ are, respectively:} \]

• A. 2, 2
• B. 2, not defined
• C. 1, 2
• D. 2, not defined

Hint:

To find degree, first make sure the equation is polynomial in derivatives. The highest exponent of the highest order derivative (after simplification) gives the degree.

Answer 1

The Correct Option is A

To determine the order and degree of a differential equation:
- The order is defined by the highest derivative present. In this equation, the highest derivative is $\frac{d^2 y}{dx^2}$, indicating an order of 2.
- The degree is the power of the highest order derivative once all fractional powers and roots have been eliminated. The term $\left( \frac{d^2 y}{dx^2} \right)^2$ is initially raised to the power of 2, resulting in an original power of 2 × 2 = 4. 
However, it is presented within the expression $\left[ \left( \frac{d^2 y}{dx^2} \right)^2 - 1 \right]^2$. 
Therefore, the degree of the highest derivative ($d^2y/dx^2$) is 2.