Using the method of undertermined coe cients solve for y giv

Using the method of undertermined coe¢ cients, solve for y given the following info

thank you.

Solution

given that y\"+y=g(t) here i sloved this prblem using equivalent equation

y\'\' + y = t/2 + u(t6) (3 t/2)
y\'\' + y = t/2 1/2 u(t6) (t 6)

Use Laplace transforms:
s² Y(s) s y(0) y\'(0) + Y(s) = 1/2 [ 1/s² e^(6s)/s * 1/s² ]
s² Y(s) 0 1 + Y(s) = 1/2 [ 1/s² e^(6s)/s * 1/s² ]
Y(s) (s² + 1) = 1/2 [ 1/s² e^(6s)/s * 1/s² ] + 1
Y(s) (s² + 1) = 1/2 [ (2s²+1)/s² e^(6s)/s * 1/s² ]
Y(s) = 1/2 [ (2s²+1)/(s²(s²+1)) e^(6s)/s * 1/(s²(s²+1)) ]

Using partial fractions, we get:
Y(s) = 1/2 [ 1/s² + 1/(s²+1) e^(6s)/s * (1/s² 1/(s²+1)) ]

Transform is
y(t) = 1/2 [ t + sin(t) u(t6) ((t6) sin(t6)) ]
y(t) = 1/2 t + 1/2 sin(t) 1/2 u(t6) (t 6 sin(t6)

here i got the solution in sin same for coe