inverse Laplace transform of this function

hi

can any body help me to compute inverse laplace transform of this function 1/(s^3+s^2+a) ,in which a is a parameter? I guess that the responce is sin(at)^2 or somthing like that,but when I use MATLAB command ilaplace it gives a long and wierd answre which seems diffrent from the mentiond one?

Answer 1

[Walter Roberson]

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))))