For the solution to the differential equation y y 2 sin

For the solution to the differential equation y \'\'\' - y \' = 2 sin t What is the coefficient of the term involving sin t ?

Solution

Let \'s substitute A sin(t) +B cos(t) to find A

(A sin(t))\'\'\'-(A sin(t)+B cos(t))\'=2 sin(t)

-(A sin(t)+B cos(t))\'-(A sin(t)+B cos(t))\'=2 sin(t)

-2(A cos(t)-B sin(t))=2 sin(t)

Hence, A=0, B=1

So particular solution is

cos(t)

Hence coefficient of sin(t) is 0