General solution of differential equation of f(x)dydx=f2(x)+f(x)y+f′(x)y (Where c being arbitrary constant)

dydx−1+f′(x)f(x)y=f(x), I.F.=e−∫1+f′(x)f(x)dx=e−xf(x)ye−xf(x)=∫e−xdx+C⇒ye−xf(x)=−e−x+C

General solution of differential equation of f(x)dydx=f2(x)+f(x)y+f′(x)y (Where c being arbitrary constant)

dydx−1+f′(x)f(x)y=f(x), I.F.=e−∫1+f′(x)f(x)dx=e−xf(x)ye−xf(x)=∫e−xdx+C⇒ye−xf(x)=−e−x+C

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General solution of differential equation of $f (x) \frac{dy}{dx} = f^{2} (x) + f (x) y + f^{'} (x) y$ (Where c being arbitrary constant)

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$y = f (x) + {ce}^{x}$

$y = cf (x) + e^{x}$

$y = - f (x) + {ce}^{x}$

$y = - f (x) + {ce}^{x} f (x)$

answer is C.

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Detailed Solution

$\frac{dy}{dx} - (1 + \frac{f^{'} (x)}{f (x)}) y = f (x), I . F . = e^{- \int (1 + \frac{f^{'} (x)}{f (x)}) dx} = \frac{e^{- x}}{f (x)}$

$\frac{{ye}^{- x}}{f (x)} = \int e^{- x} dx + C \Rightarrow \frac{{ye}^{- x}}{f (x)} = - e^{- x} + C$

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General solution of differential equation of f(x)dydx=f2(x)+f(x)y+f′(x)y (Where c being arbitrary constant)

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General solution of differential equation of $f (x) \frac{dy}{dx} = f^{2} (x) + f (x) y + f^{'} (x) y$ (Where c being arbitrary constant)