Question

Solution of the equation Xdy = $\left( y + x\frac{ƒ(y/x)}{ƒ'(y/x)} \right)$dx is –

• A. $ƒ\left( \frac{x}{y} \right)$= cy
• B. $ƒ\left( \frac{y}{x} \right)$= cx
• C. $ƒ\left( \frac{y}{x} \right)$ = cxy
• D. None of these

Answer 1

Answer: (B)

$ƒ\left( \frac{y}{x} \right)$= cx

Answer 2

We have, x dy = $\left( y + \frac{xƒ(y/x)}{ƒ'(y/x)} \right)$dx

Ž $\frac{dy}{dx}$ =$\frac{y}{x}$ + $\frac{ƒ(y/x)}{ƒ'(y/x)}$ which is homogeneous.

Put y = Vx Ž $\frac{dy}{dx}$ = V + x $\frac{dV}{dx}$,

We obtain

V + x $\frac{dV}{dx}$ = V + $\frac{ƒ(V)}{ƒ'(V)}$ d V

Ž $\frac{ƒ'(V)}{ƒ(V)}$ dV = $\frac{dx}{x}$

Integrating, we get

Ž log ƒ (V) = log cx Ž ƒ$\left( \frac{y}{x} \right)$ = cx.

Hence (2) is the correct answer