Begin by recognized this is a second order linear differencial equation with constant coefficients.

Solve, the homogenous equations,

y''+4t=0

The characheteris equation is,

k^2+4=0

Complex solutions,

k=+/- 2i

Thus the general solution is,

y=C*cos(2t)+K*sin(2t)

Here is the trick, youcannotuse the method of undetermined coefficients and look for a particular solution of the form,

y=A*sin(2t)+B*cos(2t)

For that is included in the general solution, thus, multiply by 't' and the theory gaurenttes a solution of the form,

y=t[A*sin(2t)+B*cos(2t)]