Solve using laplace transforms where U(t) is the heaviside step function

y" + 10y' + 25y = 3U(t-1) , y(0) = 0 , y' (0) =1

Apply L to both sides:
[s^2 Y(s) - 0s - 1] + 10 [s Y(s) - 0] + 25 Y(s) = 3 * e^(-1s)/s

Solve for Y(s):
(s^2 + 10s + 25) Y(s) = 1 + 3e^(-s)/s
==> Y(s) = 1/(s+5)^2 + 3e^(-s)/(s(s+5)^2)
..............= 1/(s+5)^2 + (3/25) e^(-s) [1/s - 1/(s+5) - 5/(s+5)^2], by partial fractions.

Inverting yields
y(t) = te^(-5t) + (3/25) [1 - e^(-5(t - 1)) - 5(t - 1)e^(-5(t - 1))] u(t - 1).

I hope this helps!