Question

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

• A. $f(x)=y+C$
• B. $f(x)=y(x+C)$
• C. $f(x)=x+C$
• D. None of the above

Answer 1

Answer: (B)

The given equation is
$\frac{dy}{dx}=\frac{y{f}^{\prime }(x)-y^2}{f(x)}$
$\Rightarrow y{f}^{\prime }(x)dx-f(x)dy=y^{2}dx$
$\Rightarrow \frac{y{f}^{\prime }(x)dx-f(x)dy}{y^2}=dx$
$\Rightarrow d\left\{\frac{f(x)}{y}\right\}=dx$
On integration, we get
$\frac{f(x)}{y}=x+C$
$\Rightarrow f(x)=y(x+C)$