Question

The general solution of the differential equation \(xy(y + 2y') + (y^2 - y) \, dx = 0\) is?

• A. \(\log |x + y| + \frac{2}{3} \tan^{-1} \left(\frac{y - x}{\sqrt{3} x}\right) = c\)
• B. \(\log |2x + y| + \frac{2}{3} \tan^{-1} \left(\frac{y - x}{\sqrt{3} x}\right) = c\)
• C. \(\log |x + y| + \frac{2}{3} \tan^{-1} \left(\frac{y - x}{\sqrt{3} y}\right) = c\)
• D. \(\log |x + 2y| + \frac{2}{3} \tan^{-1} \left(\frac{2y - x}{\sqrt{3} x}\right) = c\)

Hint:

Use standard methods like integrating factors or substitution to solve first-order linear differential equations.

Answer 1

The Correct Option is A

This differential equation can be solved using standard methods of solving first-order linear equations. First, rearrange and simplify the terms. The general solution for the given equation comes out to be: \[ \log |x + y| + \frac{2}{3} \tan^{-1} \left(\frac{y - x}{\sqrt{3} x}\right) = c. \]