Find general solutjion of the given differential equation y

Find general solutjion of the given differential equation:

y\'\' +2y\' +5y=3sin(2t)

Solution

The given differential equation is

y\'\' +2y\' +5y=3sin(2t)

First we find the general solution of the homogeneous equation, we have

y = c1 e^-t cos 2t + c2 e^-t sin 2t

Next we find a special solution, let Y = A cos 2t + B sin 2t, plug Y in the equation, we have

Y = 3/17 sin(2t) - 12/17 cos(2t)

Hence the general solution is

y = c1 e^-t cos 2t + c2 e^-t sin 2t + 3/17 sin(2t) - 12/17 cos(2t)