Question

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

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

Answer 1

The correct option is B $f\left(x\right)=y(x+c)$

$\frac{dy}{dx}=\frac{y{f}^{\prime }\left(x\right)-y^2}{f\left(x\right)}$
$\Rightarrow y{f}^{\prime }\left(x\right)dx-f\left(x\right)dy=y^{2}dx$
$\Rightarrow \frac{y{f}^{\prime }\left(x\right)dx-f\left(x\right)dy}{y^2}=dx$
$\Rightarrow d\left[\frac{f\left(x\right)}{y}\right]=d\left(x\right)$
Integrating both sides, we get
$\frac{f\left(x\right)}{y}=x+c$
$\Rightarrow f\left(x\right)=y(x+c)$