Answer
$\;\;\;\;\;\;C_{1}=0\;\;\;\;\;\;\;C_{2}=2\;\;\;\;\;\;\;\;\;C_{3}=1$
$y(t)=2cos(t)+sin(t)+ln[sec(t)+tan(t)]-tcos(t)+sin(t)ln[cos(t)]$
Work Step by Step
The solution of the differential equation was found out in problem (4) :
$y(t)=C_{1}+C_{2}cos(t)+C_{3}sin(t)+ln(sec(t)+tan(t))-tcos(t)+sin(t)ln(cos(t))$
${y}'=-C_{2}sin(t)+C_{3}cos(t)+tsin(t)-cos(t)-sin(t)tan(t)+cos(t)ln(cos(t))+sec(t)$
${y}''=-C_{2}cos(t)-C_{3}sin(t)+sin(t)+tcos(t)-cos(t)tan(t)-sin(t)sec^2(t)-sin(t)ln(cos(t))-sec(t)tan(t)$
$y(0)=C_{1}+C_{2}=2$
${y}'(0)=C_{3}-1+1=1$
${y}''(0)=-C_{2}=-2$
So; $\;\;\;\;\;\;C_{1}=0\;\;\;\;\;\;\;C_{2}=2\;\;\;\;\;\;\;\;\;C_{3}=1$
$y(t)=2cos(t)+sin(t)+ln(sec(t)+tan(t))-tcos(t)+sin(t)ln(cos(t))$
