Chapter 1 - First-Order Differential Equations - 1.2 Basic Ideas and Terminology - Problems - Page 22: 38

$y=-\cos x+x+1$

Integrate to turn $y''$ into $y'$. $$y''=\cos x$$ $$y'=-\sin x +C_1$$ Integrate once again to turn $y'$ into $y$. $$y=-\cos x+C_1x+C_2$$ Find the values of $y$ and $y'$ at $x=$ and match them with the conditions. $$y(0)=2=-\cos(0)+C_1(0)+C_2=1+C_2$$ $$2=1+C_2$$ $$C_2=1$$ $$y'(0)=1=-\sin(0)+C_1=C_1$$ $$C_1=1$$ Substituting these values yields $y=-\cos x+x+1$.