Question

find inverse laplase (s+5)/({s}^{2} -6 s + 13)​

Answer 1

Answer❀✿°᭄

This problem can also be done without resorting to complex numbers.

Complete the square on the denominator:

s2 + 2s + 5 = (s+1)2 + 4

Since the denominator is now expressed in terms of s+1, express the numerator the same way:

2s + 2 = 2(s+1)

Now the whole fraction is in terms of s+1. A Laplacian translation theorem says we can substitute "s" for "s+1" if we compensate by multiplying the inverse Laplacian by e-t:

f(t) = L-1{2(s+1)/[(s+1)2 + 4]}

= e-t L-1{2s/(s2 + 4)}

And the rest is easy:

f(t) = 2e-t L-1{s/(s2 + 4)}

= 2e-t cos(2t)