General solution of differential equation of f(x) (d y/d x)=f2(x)+f(x) y+f prime(x) y is: ( c being arbitrary constant).

Question

General solution of differential equation of f(x) (d y/d x)=f2(x)+f(x) y+f prime(x) y is: ( c being arbitrary constant). (A) y=f(x)+c ex (B) y=-f(x)+c ex (C) y=-f(x)+c ex f(x) (D) y=c f(x)+ex

Tardigrade Admin · Accepted Answer

(d y/d x)-(1+(f prime(x)/f(x))) y=f(x) IE .=e-∫(1+(f prime(x)/f(x))) d x=(e-x/f(x)) (y e-x/f(x))=∫ e-x d x+C ⇒ (y e-x/f(x))=-e-x+C

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

$y = f (x) + c e^{x}$

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

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

$y = c f (x) + e^{x}$

$\frac{d y}{d x} - (1 + \frac{f ^{'} ( x )}{f ( x )}) y = f (x)$
$I E . = e^{- \int (1 + \frac{f ^{'} ( x )}{f ( x )}) d x} = \frac{e ^{- x}}{f ( x )}$
$\frac{y e ^{- x}}{f ( x )} = \int e^{- x} d x + C$
$\Rightarrow \frac{y e ^{- x}}{f ( x )} = - e^{- x} + C$

Q. General solution of differential equation of f(x)dxdy​=f2(x)+f(x)y+f′(x)y is : ( c being arbitrary constant).

y=f(x)+cex

y=−f(x)+cex

y=−f(x)+cexf(x)

y=cf(x)+ex