Question

Let \(f\) be a differentiable function with \(\lim _{x \rightarrow \infty} f(x)=0\). If \(y^{\prime}+y f^{\prime}(x)-f(x) f^{\prime}(x)=0, \lim _{x \rightarrow \infty} y(x)=0\), then \(\left(\right.\) where \(\left.y^{\prime}=\frac{d y}{d x}\right)\)
(A) \(y+1=e^{f(x)}+f(x)\)
(B) \(y-1=e^{f(x)}+f(x)\)
(C) \(y+1=e^{-f(x)}+f(x)\)
(D) \(y-1=e^{-10 x}+f(x)\)

Answer 1

Ans: (C)
Hint : \(\frac{d y}{d x}+f^{\prime}(x) y=f^{\prime}(x) f(x)\)
$$
\begin{aligned}
&\Rightarrow y \times e^{f(x)}=\int f^{\prime}(x) f(x) e^{f(x)} d x \\
&\left.\Rightarrow y \times e^{f(x)}=e^{f(x)}(f(x)-1)+c \quad \text { [Putting } f(x)=0 ; y=0, c=1\right] \\
&\Rightarrow y \times e^{f(x)}=e^{f(x)}(f(x)-1)+1 \\
&\Rightarrow y=f(x)-1+e^{-f(x)} \\
&\Rightarrow y+1=e^{-f(x)}+f(x)
\end{aligned}
$$

Answer 2

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