Q5Advanced Engineering Mathematics I

Question

Find Inverse Laplace transform of $\frac{S}{S ^{4} + 4 a ^{4}}$ .

Answer 1

Given $F (s) = \frac{s}{s ^{4} + 4 a ^{4}}$ .

We can factor the denominator by completing the square: $s^{4} + 4 a^{4} = (s^{2} + 2 a^{2})^{2} - 4 a^{2} s^{2} = (s^{2} - 2 a s + 2 a^{2}) (s^{2} + 2 a s + 2 a^{2})$

Using partial fractions: $\frac{s}{s ^{4} + 4 a ^{4}} = \frac{1}{4 a ^{2}} [\frac{s}{s ^{2} - 2 a s + 2 a ^{2}} - \frac{s}{s ^{2} + 2 a s + 2 a ^{2}}]$

Each term can then be inverted using the shifting theorem, resulting in combinations of $e^{a t} sin a t$ and $e^{a t} cos a t$ .

Q5Advanced Engineering Mathematics I

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