Question

The general solution of $\frac{dy}{dx}$ = $\frac{2x - y}{x + 2y}$is –

• A. x² – xy + y² = c
• B. x² – xy – y² = c
• C. x² + xy – y² = c
• D. x² + xy² = c

Answer 1

Answer: (B)

x² – xy – y² = c

Answer 2

Given that,$\frac{dy}{dx}$ = $\frac{2x - y}{x + 2y}$ ... (i)

Let y = vx Ž $\frac{dy}{dx}$ = v + x $\frac{dv}{dx}$

\ v + x$\frac{dv}{dx}$ = $\frac{2 - v}{1 + 2v}$

Ž x$\frac{dv}{dx}$= $\frac{2 - v - v(1 + 2v)}{1 + 2v}$

Ž $\int_{}^{}\frac{1 + 2v}{2(1 - v - v^{2})}$dv = $\int_{}^{}\frac{1}{x}$dx

Ž log k – $\frac{1}{2}$log (1 – v – v²) = log x

Ž 2 log k – log (1 – v – v²) = 2 log x

Ž log c = log [x² (1 – v – v²)]

Ž c = x² $\left( 1 - \frac{y}{x} - \frac{y^{2}}{x^{2}} \right)$

Ž x² – xy – y² = c