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2902 Laplace Inverse by Table Lookup

The first thing we need to be able to do is to use the Laplace table to find the inverse Laplace transform. We will illustrate this entirely by examples.

Notation: The inverse Laplace transform will be denoted L1.

Example 1. Find L1(1/(s2)).
Solution. Use the table entry L(eat)=1/(sa): L1(1/(s2))=e2t

Example 2. Find L1(1/(s2+9)).
Solution. Use the table entry L(sin(ωt))=ω/(s2+ω2) and linearity: L1(1/(s2+9))=13L1(3/(s2+9))=13sin(3t)

Example 3. Find L1(4/s2).
Solution. Use the table entry L(t)=1/s2: L1(4/s2)=4t.

Example 4. Find L1(4/(s2)2).
Solution. Use the s-shift formula L(eztf(t))=F(s2), where, in this case, F(s)=4/s2\rArrf(t)=4t by example (3).
Therefore, L1(4/(s2)2)=L1(F(s2))=e2tf(t)=e2t4t.

Example 5. Find L1(1s2+4s+13).
Solution. We first need to complete the square s2+4s+13=s2+4s+4+9=(s+2)2+9 We have a shifted function F(s+2), where F(s)=1/(s2+9). Using example (2), we know that f(t)=sin(3t)/3, so using the s-shift rule we get L1(1s2+4s+13)=L1(F(s+2))=e2tf(t)=e2tsin(3t)3

Example 6. Find L1(s(s2+ω2)2).
Solution. We haven't seen this formula yet, but there is a table entry, which gives: t2ωsin(ωt).

Example 7. Find L1(1(s2+ω2)2).
Solution. This is also a table entry, answer: 12ω3(sin(ωt)ωtcos(ωt)).