Sorry, the answer isn't even close to sin(a*t)^2. It really is a long weird answer.
syms t
alpha = roots([1 1 0 a]); %3 roots of cubic
tot = 0;
for K = 1 : length(alpha)
tot = tot + exp(alpha(K)*t) / (alpha(K) * (3 * alpha(K) + 2));
end
Or as Maple would write it,
sum(exp(_alpha*t)/_alpha / (3*_alpha+2), _alpha = RootOf(_Z^3+_Z^2+a))
Or if you prefer the explicit solution,
exp(((1/6)*(-8 - 108*a + 12*(12*a+81*a^2)^(1/2))^(1/3) + (2/3)/(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) - 1/3)*t)/(((1/6)*(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) + (2/3)/(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) - 1/3)*((1/2)*(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) + 2/(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) + 1)) +
exp((-(1/12)*(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) - (1/3)/(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) - 1/3+((1/2)*I)*3^(1/2)*((1/6)*(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) - (2/3)/(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3)))*t)/((-(1/12)*(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) - (1/3)/(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) - 1/3 + ((1/2)*I)*3^(1/2)*((1/6)*(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) - (2/3)/(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3)))*(-(1/4)*(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) - 1/(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) + 1 + ((3/2)*I)*3^(1/2)*((1/6)*(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) - (2/3)/(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3)))) +
exp((-(1/12)*(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) - (1/3)/(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) - 1/3 - ((1/2)*I)*3^(1/2)*((1/6)*(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) - (2/3)/(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3)))*t)/((-(1/12)*(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) - (1/3)/(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) - 1/3 - ((1/2)*I)*3^(1/2)*((1/6)*(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) - (2/3)/(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3)))*(-(1/4)*(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) - 1/(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) + 1 - ((3/2)*I)*3^(1/2)*((1/6)*(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3) - (2/3)/(-8 - 108*a + 12*(12*a + 81*a^2)^(1/2))^(1/3))))
