Use appropriate algebra etc. to find : (inverse Laplace transform)?

Question

L^(-1){1 / (s^(2) + s - 20)}

Thanks!

Answer 1

1 / s^2 + s - 20
= 1 / ((s-4) (s+5))
Use partial fractions to split this into separate terms:
Let 1 / ((s-4)(s+5)) = A / (s-4) + B / (s+5)
Then A (s+5) + B (s-4) = 1 for all s
so (A+B) s + (5A-4B) = 1 for all s
so A + B = 0 and 5A - 4B = 5A + 4A = 1, so A = 1/9 and B = -1/9.

So L^(-1){1 / (s^2 + s - 20)}
= L^(-1){(1/9) (1/(s-4)} + L^(-1){(-1/9) 1/(s+5)}
= (1/9) e^(4t) + (-1/9) e^(-5t)
= (e^4t - e^(-5t)) / 9.