Question

If $f\left(x\right)=f^{\prime }\left(x\right)$ and $f\left(0\right)=3$, and $f\left(2\right)$ equals

• A. $4e^3$
• B. $3e^4$
• C. $2e^3$
• D. $3e^2$

Answer 1

The correct option is D $3e^2$

Given $f\left(x\right)=f^{\prime }\left(x\right)$
which can be written as
$y=\frac{dy}{dx}$
$\Rightarrow \frac{dy}{dx}-y=0$
which is a linear differential eqn.
Here, $P=-1,Q=0$
Integrating factor $I.F=e^{\int -1dx}$
$\Rightarrow I.F.=e^{-x}$
Solution is given by
$y{e}^{-x}=\int 0e^{-x}dx$
$\Rightarrow y{e}^{-x}=c$ ....(1)
Since, $f\left(0\right)=3$
$\Rightarrow 3e^0=c$
$\Rightarrow c=3$
Substitute this value in (1), we get
$y{e}^{-x}=3$
Substitute $x=2$
$\Rightarrow y{e}^{-2}=3$
$\Rightarrow y=f\left(2\right)=3e^2$