Topic: Engineering Mathematics
Laplace transform of $t \cos(at)$ is
- A. $\frac{s^{2}+a^{2}}{(s^{2}-a^{2})^{2}}$
- B. $\frac{s}{(s^{2}-a^{2})^{2}}$
- C. $\frac{s^{2}-a^{2}}{(s^{2}+a^{2})^{2}}$
- D. $\frac{s}{(s^{2}+a^{2})^{2}}$
Answer: C. $\frac{s^{2}-a^{2}}{(s^{2}+a^{2})^{2}}$
Explanation: By the property of Laplace transforms, $L[t^n f(t)] = (-1)^n \frac{d^n}{ds^n}[F(s)]$. Here, $f(t) = \cos(at)$ and $F(s) = \frac{s}{s^2+a^2}$. For $n=1$, $L[t \cos(at)] = -\frac{d}{ds}\left(\frac{s}{s^2+a^2}\right)$. Using the quotient rule: $-\left[\frac{(s^2+a^2)\cdot 1 - s \cdot (2s)}{(s^2+a^2)^2}\right] = -\left[\frac{a^2-s^2}{(s^2+a^2)^2}\right] = \frac{s^2-a^2}{(s^2+a^2)^2}$.