Question

The solution of the differential equation $\frac{dy}{dx}=\frac{y{\varphi }^{\prime }\left(x\right)-y^2}{\varphi \left(x\right)}$ is

• A. $y=\frac{x}{\varphi \left(x\right)+c}$
• B. $y=\frac{x}{\varphi \left(x\right)}+c$
• C. $y=\frac{\varphi \left(x\right)+c}{x}$
• D. $y=\frac{\varphi \left(x\right)}{x+c}$

Answer 1

The correct option is D $y=\frac{\varphi \left(x\right)}{x+c}$

Given

$y^{\prime }=\frac{y{\varphi }^{\prime }\left(x\right)-y^2}{\varphi \left(x\right)}$

$\Rightarrow y^{\prime }\varphi \left(x\right)=y{\varphi }^{\prime }\left(x\right)-y^2$

$\Rightarrow y^2=y{\varphi }^{\prime }\left(x\right)-y^{\prime }\varphi \left(x\right)$

$\Rightarrow \frac{y{\varphi }^{\prime }\left(x\right)-y^{\prime }\varphi \left(x\right)}{y^2}=1$

$\Rightarrow {\left(\frac{\varphi \left(x\right)}{y}\right)}^{\prime }=1$

$\Rightarrow \frac{\varphi \left(x\right)}{y}=x+c$

$\Rightarrow y=\frac{\varphi \left(x\right)}{x+c}$