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)

$f(x) = y(x+C)$

Answer 2

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 $\frac{f(x)}{y}=x+C$ $\Rightarrow f(x)=y(x+C)$