Question

The solution of the differential equation
$\frac{dy}{dx}=\frac{yf '\left(x\right)-y^{2}}{f \left(x\right)}$ 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{d y}{d x}=\frac{y f^{\prime}(x)-y^{2}}{f(x)} $
$\Rightarrow y f^{\prime}(x) d x-f(x) d y=y^{2} d x $
$\Rightarrow \frac{y f^{\prime}(x) d x-f(x) d y}{y^{2}}=d x $
$\Rightarrow d\left\{\frac{f(x)}{y}\right\}=d x$
On integration, we get
$ \frac{f(x)}{y}=x+C $
$\Rightarrow f(x)=y(x+C)$