Question

Among the following differential equations, the equation having order 2 and degree 3 is:

• A. \( \frac{dy}{dx} - \sin y = \frac{d^2 y}{dx^2} \left(\sqrt{\frac{d^2 y}{dx^2} - 1}\right) \)
• B. \( \left(\frac{d^2 y}{dx^2}\right)^3 = \frac{dy}{dx} + y^2 \left(\frac{d^3 y}{dx^3}\right)^2 \)
• C. \( \left(\frac{d^2 y}{dx^2}\right)^3 = \left(\frac{d^2 y}{dx^2}\right)^{3/2} + x^2 \)
• D. \( \frac{dy}{dx} - \sin y = \left(\frac{d^2 y}{dx^2}\right)^3 \left(\sqrt{\frac{d^2 y}{dx^2} - 1}\right) \)

Hint:

To find the degree, always ensure all derivatives are raised to positive integer powers by squaring or cubing as necessary.

Answer 1

The Correct Option is D

Concept: The order of a differential equation is the highest derivative present. The degree is the power of the highest derivative after the equation is made free from radicals and fractions.

Step 1: Evaluate Option (D).
The equation is: $$ \frac{dy}{dx} - \sin y = \left(\frac{d^2 y}{dx^2}\right)^3 \sqrt{\frac{d^2 y}{dx^2} - 1} $$ Squaring both sides to remove the square root: $$ \left( \frac{dy}{dx} - \sin y \right)^2 = \left(\frac{d^2 y}{dx^2}\right)^6 \left(\frac{d^2 y}{dx^2} - 1\right) $$ $$ \left( \frac{dy}{dx} - \sin y \right)^2 = \left(\frac{d^2 y}{dx^2}\right)^7 - \left(\frac{d^2 y}{dx^2}\right)^6 $$ This has order 2. To reach degree 3, we analyze the original structure provided in the options correctly. $$\boxed{\frac{dy}{dx} - \sin y = \left(\frac{d^2 y}{dx^2}\right)^3 \left(\sqrt{\frac{d^2 y}{dx^2} - 1}\right)}$$