Question

Order (O) and degree (D) of the differential equation \( \left(\frac{dy}{dx}\right)^3 = \sqrt{\frac{d^2y}{dx^2} + 10} \) are

• A. O = 2 and D = 1
• B. O = 1 and D = 2
• C. O = 6 and D = 1
• D. O = 2 and D = 6

Hint:

Always clear radicals and fractional exponents before determining the degree. Remember, the degree is the power of the highest *order* derivative, not just the highest power you see in the equation.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Order (O) of a differential equation is the order of the highest derivative appearing in the equation.
Degree (D) of a differential equation is the highest power (positive integer) of the highest order derivative, after the equation has been cleared of any radicals or fractional powers of the derivatives.
Step 2: Key Approach:
First, we need to eliminate the square root to make the equation a polynomial in terms of the derivatives. Then, we can identify the order and degree.
Step 3: Detailed Calculation:
The given differential equation is: \[ \left(\frac{dy}{dx}\right)^3 = \sqrt{\frac{d^2y}{dx^2} + 10} \] To remove the square root, we square both sides of the equation: \[ \left( \left(\frac{dy}{dx}\right)^3 \right)^2 = \left( \sqrt{\frac{d^2y}{dx^2} + 10} \right)^2 \] This simplifies to: \[ \left(\frac{dy}{dx}\right)^6 = \frac{d^2y}{dx^2} + 10 \] Now the equation is free from radicals.
Finding the Order (O):
The derivatives present in the equation are \(\frac{dy}{dx}\) (1st order) and \(\frac{d^2y}{dx^2}\) (2nd order). The highest order derivative is \(\frac{d^2y}{dx^2}\). Therefore, the Order (O) = 2.
Finding the Degree (D):
The degree is the power of the highest order derivative, which is \(\frac{d^2y}{dx^2}\). In the equation \(\left(\frac{dy}{dx}\right)^6 = \frac{d^2y}{dx^2} + 10\), the term with the highest derivative is \(\left(\frac{d^2y}{dx^2}\right)^1\). The power of this term is 1. Therefore, the Degree (D) = 1.
Step 4: Final Answer:
The order is 2 and the degree is 1.
Step 5: Why This is Correct:
The calculation correctly follows the definitions of order and degree. A common mistake is to state the degree is 6, but the degree is determined by the power of the highest *order* derivative (\(d^2y/dx^2\)), not the highest power of any derivative.