Question

The general solution of the differential equation \[ \frac{dy}{dx} = \frac{2x^2 - xy - y^2}{x^2 - y^2} \]

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

Hint:

When solving nonlinear differential equations, try factoring and checking for opportunities to separate variables or use substitution that simplifies the rational expression.

Answer 1

The Correct Option is B

Given: \[ \frac{dy}{dx} = \frac{2x^2 - xy - y^2}{x^2 - y^2} \] Step 1: Rewrite the RHS: \[ \frac{2x^2 - xy - y^2}{x^2 - y^2} = \frac{(2x^2 - y^2) - xy}{x^2 - y^2} = \frac{2x^2 - y^2}{x^2 - y^2} - \frac{xy}{x^2 - y^2} \] Step 2: Solve using variable substitution and method of integration (skipping intermediate integration for brevity). After integrating and simplifying, the general solution is obtained as: \[ \boxed{ \sqrt{2} \log \left| \frac{y^2 - 2x^2}{x^2} \right| + \log \left| \frac{y - \sqrt{2}x}{y + \sqrt{2}x} \right| + 2\sqrt{2} \log |x| = c } \]