How to solve this Laplace transform? $f(t)=e^{-2t}\cos^2 3t - 3t^2 e^{3t}$

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Remember that

$$\cos^2(3t) = \frac{1+\cos(6t)}{2}$$

and

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

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Updated on October 01, 2020

Comments

  • Admin
    Admin about 3 years

    Find the laplace transform of

    $$f(t)=e^{-2t}\cos^2 3t - 3t^2 e^{3t}$$

    The answer is $$\frac{1}{2(s+2)}+ \frac{1}{2} \frac{s+2}{s^2 + 4s + 40} - \frac{6}{(s-3)^3}.$$

    This took me about an hour to solve, which seems ridiculously long. I probably did things inefficiently, how can this be solved?

    My method: Separate using linearity and then integrate by parts, with partial fraction decomposition.

    • Admin
      Admin about 9 years
      Edited to be objective.(in reference to the close vote)
    • JP McCarthy
      JP McCarthy about 9 years
      You must not have done the shift theorem yet... this will make things much easier for you.