This is a linear ordinary differential equation (since you have the explicit form for u(t)).
Try using Differential Operator techniques. Let P(D) = (D^4 + 6D^2 + 25). You will need to solve 1/P(D)[u' + 3u] for the specific solution p_s and solve P(D)y = 0 using the substitution y = e^(mx) for the complementary solution p_c.
Then your solution will be the sum of these giving y(t) = p_c + p_s (which are both functions of t).