Question

The value of $g\left(0\right)-f\left(0\right)$ equals

• A. $\frac{2}{3-e^2}$
• B. $\frac{3}{e^2-2}$
• C. $\frac{1}{e^2-1}$
• D. $0$

Answer 1

The correct option is A $\frac{2}{3-e^2}$

$g\left(x\right)={\int }_0^1{e}^{x+t}.g\left(t\right)dt+x$

$g\left(x\right)=e^x({\int }_0^1{e}^t.g\left(t\right)dt)+x$

$g\left(x\right)=k{e}^x+x$

Again by definition

$k=({\int }_0^1{e}^t.g\left(t\right)dt)$

$k=({\int }_0^1{e}^x(k{e}^x+x)dx)$

Integrate using by parts to get

$k=(k\frac{e^{2x}}{2}+x{e}^x-e^x){|}_0^1$

Put the limits and solve to get

$k=\frac{2}{3-e^2}$.

.

$g\left(0\right)-f\left(0\right)=\frac{2}{3-e^2}$