Question

Find the function which has the following Laplace transform

4/(S^2(S^2+4))

Answer 1

\[4/(s ^{2}(s ^{2}+4) = a /s ^{2} + b/(s ^{2}+4)\]\[=(a*(s ^{2}+4)+b*s ^{2})/(s ^{2}*(s ^{2}+4))\]\[=((a+b)*s ^{2} + 4*a)/(s ^{2}(s ^{2}+4))\]so by comparing the nominators of the first and last expressions
a+b=0 and 4*a=4
so a = 1 and b = -1\[4/(s ^{2}(s ^{2}+4))= 1/s ^{2}-1/(s ^{2}+4)\]

Answer 2

so 1/s^2 is the laplace transform of t.u(t) (ramp function)

Answer 3

as for 1/(s^2+4)
\[1/(s^2+4)=(1/2) * 2/(s^2+2^2)\]which is the laplace transform of\[(1/2)*\sin (2t) . u(t)\]

Answer 4

finally the function we are looking for is f(t)=t.u(t) - (1/2)*sin (2t) . u(t)
f(t) = (t-(1/2)*sin (2t))*u(t)

Answer 5

That's brilliant man, thanks!!!!