Question

The solution of differential equation $yy^{'} = x\left( \frac{y^{2}}{x^{2}} + \frac{\varphi(y^{2}/x^{2})}{\varphi^{'}(y^{2}/x^{2})} \right)$ is

• A. $\varphi(y^{2}/x^{2}) = cx^{2}$
• B. $x^{2}\varphi(y^{2}/x^{2}) = c^{2}y^{2}$
• C. $x^{2}\varphi(y^{2}/x^{2}) = c$
• D. $\varphi(y^{2}/x^{2}) = \frac{cy}{x}$

Answer 1

Answer: (A)

$\varphi(y^{2}/x^{2}) = cx^{2}$

Answer 2

Given equation may be re-written as

$\frac{y}{x} \cdot \frac{dy}{dx} = \left( \frac{y}{x} \right)^{2} + \frac{\varphi((y/x)^{2})}{\varphi^{'}((y/x)^{2})}$ .....(i)

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

∴ From (i), $v\left( v + x\frac{dv}{dx} \right) = v^{2} + \frac{\varphi(v^{2})}{\varphi^{'}(v^{2})}$ ⇒ $vx\frac{dv}{dx} = \frac{\varphi(v^{2})}{\varphi^{'}(v^{2})}$

⇒ $\frac{\varphi^{'}(v^{2})(2vdv)}{\varphi(v^{2})} = 2\frac{dx}{x}$

Integrating, $\ln(\varphi(v^{2})) = 2\ln x + \ln c$ ⇒ $\varphi(v^{2}) = cx^{2}$

∴ $\varphi(y^{2}/x^{2}) = cx^{2}$