Question

Solve the initial value problem.

y′′+y=1, y(0)=2, y′(0)=7.

Answer 1

$y''+\left( y-1 \right) =0\\\left( y-1 \right) ''+\left( y-1 \right) =0\\\lambda ^2+1=0\Rightarrow \lambda =\pm i\\y-1=C_1\cos t+C_2\sin t\\y=1+C_1\cos t+C_2\sin t\\\left\{ \begin{array}{c} y\left( 0 \right) =2\\ y'\left( 0 \right) =7\\\end{array} \right. \Rightarrow \left\{ \begin{array}{c} 1+C_1=2\\ C_2=7\\\end{array} \right. \Rightarrow \left\{ \begin{array}{c} C_1=1\\ C_2=7\\\end{array} \right. \\y=1+7\cos t+\sin t$