Can anyone help me solve this initial value problem?

Question

dy/dt + 2y = 4sint

- i found the integrating factor I(x) and multiplied through by it but i end up with

ye^(2t) = Integral of (4sint * e^2t) dt and i dont know where to go from there

i forgot to say that the initial condition is that y(0) = 1/5

Answer 1

The solution of this equation with second side = 0 is
y=Ce^-2t
To find a particular solution of the equation with 2nd side
try
y=a cost +b sin t
y´=-a sint+b cost
(2a+b)cos t +(2b-a)sin t = 4 sin t
so
2a+b=0
2b-a=4
b=-2a and -5a=4 and a =-4/5 and b=8/5
So your solution is
y=Ce^-2t - 4/5 cos t +8/5 sin t ( arbitrary constant depending on initial condition)