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    laplace

    Find f(t) if it satisfies the equation:

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    Quote Originally Posted by boousaf View Post
    Find f(t) if it satisfies the equation:

    The answers are all here. If f(t) = t^n then \mathcal{L}(f)(s) = \frac{n!}{s^{n+1}}. So the Laplace transform of 6t^3 is \frac{36}{s^4}. Also, the Laplace transform converts convolution products to regular products. So \mathcal{L}((f*f)(t)) = (\mathcal{L}(f)(s))^2. Therefore (\mathcal{L}(f)(s))^2 = \frac{36}{s^4}. Take the square root of both sides to get \mathcal{L}(f)(s), then look up the inverse Laplace transform of that to find f(t).
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