Question

General solution of differential equation of $f(x) \frac{d y}{d x}=f^{2}(x)+f(x) y+f^{\prime}(x) y$ is : ( $c$ being arbitrary constant).

• A. $y=f(x)+c e^{x}$
• B. $y=-f(x)+c e^{x}$
• C. $y=-f(x)+c e^{x} f(x)$
• D. $y=c f(x)+e^{x}$

Answer 1

Answer: (C)

$\frac{d y}{d x}-\left(1+\frac{f^{\prime}(x)}{f(x)}\right) y=f(x)$
$IE .=e^{-\int\left(1+\frac{f^{\prime}(x)}{f(x)}\right) d x}=\frac{e^{-x}}{f(x)}$
$\frac{y e^{-x}}{f(x)}=\int e^{-x} d x+C $
$\Rightarrow \frac{y e^{-x}}{f(x)}=-e^{-x}+C$