Question

Laplace transform of t^2 sin⁡(2t).

(a) \(\left [\frac{12s^2-16}{(s^2+4)^4}\right ]\)

(b) \(\left [\frac{3s^2-4}{(s^2+4)^3}\right ]\)

(c) \(\left [\frac{12s^2-16}{(s^2+4)^6}\right ]\)

(d) \(\left [\frac{12s^2-16}{(s^2+4)^3}\right ]\)

I got this question in unit test.

This interesting question is from Laplace Transform by Properties in chapter Laplace Transform of Engineering Mathematics

Answer 1

Right option is (d) \(\left [\frac{12s^2-16}{(s^2+4)^3}\right ]\)

To explain: We know that,

\(L(t^n f(t))=(-1)^n \frac{d^n F(s)}{ds^n}\),

Here, f(t)=sin⁡(2t)=>F(s)=\(\frac{2}{s^2+4}\),

Hence, \(L(t^2 sin⁡(2t))=\frac{d^2}{ds^2} (\frac{2}{s^2+4})=\frac{d}{ds} \frac{(s^2+4).0-2(2s)}{(s^2+4)^2}\)

=\(-4\left [\frac{(s^2+4)^2-2s(s^2+4)2s}{(s^2+4)^4} \right ]=\left [\frac{12s^2-16}{(s^2+4)^3}\right ]\)