Mathematics - Differential Equation Question with Solution | TestHub

$\begin{array}{rl}& y=1-2x+e^x{\int }_0^x{e}^{-t}f(t)dt\\ & \frac{dy}{dx}=-2+e^{-x}\cdot e^{x}f(x)+e^x{\int }_0^x{e}^{-t}f(t)dt\\ & \frac{dy}{dx}=-2+y+y+2x-1\\ & \frac{dy}{dx}-2y=(2x-3)\\ & y{e}^{-2x}=\int (2x-3)dx\cdot e^{-2x}\\ & y{e}^{-2x}=\frac{-(2x-3)}{2}{e}^{-2x}+\int e^{-2x}dx\\ & y{e}^{-2x}=\frac{-(2x-3)}{2}{e}^{-2x}-\frac{1}{2}{e}^{-2x}+c\\ & f(0)=1\Rightarrow c=1-\frac{3}{2}+\frac{1}{2}=0\end{array}$
$\begin{array}{rl}& y=-\frac{(2x-3)}{2}-\frac{1}{2}\\ & y=-x+1\\ & x+y=1\\ & \text{ area }=\frac{1}{2}(1)(1)=\frac{1}{2}\end{array}$