s^{2}F(s) + sF(s) - 6F(s) = (s^{2}+4)/(s^{2}+s) Solve for L^{-1}{F}
Attempt:
Simplified the equation to F(s) = (s^{2}+4)/[s*(s+1)*(s+3)*(s-2)]
Fraction Decomposition s^{2}+4 = A(s+1)(s+3)(s-2) + B*s(s+3)(s-2) + C*s(s+1)(s-2) + D*s(s+1)(s+3)
For s = 0 ----> -6A = 4 ----> A = -2/3
For s = -1 ----> 6B = 5 ----> b = 5/6
For s = -3 ----> -30C = 13 ----> C = -13/30
For s = 2 ----> 30D = 8 ----> D = 4/15
Inverses I get:
A: -2/3
B: 5/6 * e^{-t }C: -13/30 * e^{-3t}
D: 4/15 * e^{2t}
----> L^{-1}{F} = -2/3 + 5/6 * e-t + -13/30 * e^{-3t }+ 4/15 * e^{2t} where s > -3
I think I have an error in my factoring, but I am not sure. Could anyone help me out with this?