Question

The order and degree of the differential equation \[ y \, dx + x \, \log{\left(\frac{y}{x}\right)} \, dy - 2x \, dy = 0 \] are respectively:

• A. \( 1, 1 \)
• B. \( 1, 2 \)
• C. \( 2, 1 \)
• D. \( 1, {not defined} \)

Hint:

The order of a differential equation is determined by the highest derivative present, while the degree is the power of the highest derivative after removing any radicals or fractional powers.

Answer 1

The Correct Option is A

Step 1: The given differential equation is: \[ y \, dx + x \, \log{\left(\frac{y}{x}\right)} \, dy - 2x \, dy = 0 \] Rewriting it, we get: \[ y \, dx + \left( x \log{\left(\frac{y}{x}\right)} - 2x \right) dy = 0 \] 
Step 2: The equation is of the first order as the highest derivative involved is \( \frac{dy}{dx} \). 
Step 3: The equation does not involve any fractional power of the highest derivative, and the degree is the exponent of the highest derivative, which is 1.