Question

The order and degree of the differential equation \(\frac{d^2 y}{dx^2} + \sqrt{x + \left(\frac{dy}{dx}\right)^3 = 0\) is}

• A. (2,2)
• B. (3,2)
• C. (2,3)
• D. (1,3)

Hint:

Always square/rationalize before determining degree! $\left(\frac{d^2 y}{dx^2}\right)^2 = x + (y')^3 \implies \text{Order}=2, \text{Degree}=2$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The order of a differential equation is the highest derivative present. The degree is the power of the highest derivative after making the equation free from radicals and fractional exponents.
Key Formula or Approach:
\[ \text{Order} = \text{Highest order derivative present} \]
\[ \text{Degree} = \text{Power of highest derivative in polynomial form} \]

Step 2: Detailed Explanation:
Given differential equation:
\[ \frac{d^2 y}{dx^2} + \sqrt{x + \left(\frac{dy}{dx}\right)^3} = 0 \]

Step 1: Isolate the radical term:
\[ \frac{d^2 y}{dx^2} = -\sqrt{x + \left(\frac{dy}{dx}\right)^3} \]
Square both sides to eliminate the square root radical:
\[ \left( \frac{d^2 y}{dx^2} \right)^2 = x + \left( \frac{dy}{dx} \right)^3 \]
- The highest derivative involved is \(\frac{d^2 y}{dx^2}\), so Order = 2.
- The exponent/power of this highest derivative \(\left(\frac{d^2 y}{dx^2}\right)^2\) is 2, so Degree = 2.

Step 3: Final Answer:
Thus, the order and degree is (2,2), corresponding to option (A).