Question

Find the order and degree of the differential equation: \[ xy \left( \frac{d^2y}{dx^2} \right) + x \left( \frac{dy}{dx} \right)^2 - y \frac{dy}{dx} = 0 \]

• A. Order = 2, Degree = 1
• B. Order = 2, Degree = 2
• C. Order = 1, Degree = 2
• D. Order = 1, Degree = 1

Hint:

To find the order of a differential equation, look for the highest derivative. To find the degree, ensure the equation is polynomial in derivatives, and check the power of the highest derivative.

Answer 1

The Correct Option is A

To determine the order and degree of the given differential equation: \[ xy \left( \frac{d^2y}{dx^2} \right) + x \left( \frac{dy}{dx} \right)^2 - y \frac{dy}{dx} = 0 \] Order: The order of a differential equation is determined by the highest derivative present in the equation. In this equation, the highest derivative is \( \frac{d^2y}{dx^2} \), so the order is 2. Degree: The degree of a differential equation is the power of the highest order derivative, provided the equation is free from fractional or negative powers of derivatives. In this case, the highest derivative \( \frac{d^2y}{dx^2} \) is raised to the power of 1, so the degree is 1. Thus, the order is 2 and the degree is 1, and the correct option is: \[ \boxed{\text{Order = 2, Degree = 1}} \]