Question

Let f be a differentiable function such that $f^{\prime }\left(x\right)=f\left(x\right)+{\int }_0^{2}f\left(x\right)dx,f\left(0\right)=\frac{4-e^2}{3}$, then f(x) is

• A. $e^x-\left(\frac{e^2-1}{3}\right)$
• B. $e^x-\left(\frac{e^2-2}{3}\right)$
• C. $e^x+\frac{e-[arg|z-1|]}{3}$
• D. $2^{ex}-\frac{e^2-1}{\left[arg\right(-|z|\left)\right]}$ (where [.] greatest integer & z is a complex number)

Answer 1

The correct option is A $e^x-\left(\frac{e^2-1}{3}\right)$

$f^{\prime }\left(x\right)=f\left(x\right)+k\int \frac{f^{\prime }\left(x\right)}{f\left(x\right)+k}=\int dxlog\left(f\right(x)+k)=x+cf\left(x\right)=k_1{e}^x-kf\left(0\right)=k_1-k=\frac{4-e^2}{3}$ ....(1)
$k={\int }_0^{2}f\left(x\right)dx3k=k_1(e^2-1)$ ....(2)
Solving (1) and (2), we get
$f\left(x\right)=e^x-\left(\frac{e^2-1}{3}\right)$