Chapter 1 - First-Order Differential Equations - 1.2 Basic Ideas and Terminology - Problems - Page 22: 42

$y=e^{-x}-e^{-1}x$

Integrate the function to turn $y''$ into $y'$. $$y''=e^{-x}$$ $$y'=-e^{-x}+C_1$$ Integrate once again to turn $y'$ into $y$. $$y=e^{-x}+C_1x+C_2$$ Solve for the initial values. $$y(0)=e^{-0}+C_1(0)+C_2=1+C_2$$ $$1+C_2=y(0)=1$$ $$C_2=0$$ $$y(1)=e^{-1}+C_1+C_2$$ $$0=e^{-1}+C_1+C_2$$ $$C_1=-e^{-1}$$ Substitute these constant values to get $y=e^{-x}-e^{-1}x$.