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I am trying to follow this discussion of Laplace transforms on youtube: https://www.youtube.com/watch?v=ofvkZXgbIxE&t=610s

The relevant portion is 10 minutes in to the video.

There is some algebra that I did not completely follow. Can someone help me understand why the last step is valid? Maybe we can break it into more steps, or describe the underlying technique used? Thanks !

UPDATE: Seems like the last step was done by partial fraction decomposition. But I'm not seeing it ;^(

$$ -y(0) + (s+1) \int_{0}^\infty y e^{-st} \, dt = \frac{4}{s-3} $$

$$ (s+1) \int_{0}^\infty y e^{-st} \, dt = y(0) + \frac{4}{s-3} $$

$$ \int_{0}^\infty y e^{-st} \, dt = \frac{y(0)}{s+1} + \frac{4}{(s-3)(s+1)} $$

$$ \int_{0}^\infty y e^{-st} \, dt = \frac{y(0) - 1}{s+1} + \frac{1}{s-3}. $$

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    $\begingroup$ there is a typo: the numerator 4 in the last equation should be 1, because $\frac{1}{s-3}-\frac{1}{s+1}=\frac{4}{(s-3)(s+1)}$. $\endgroup$
    – Carlo Beenakker
    Commented yesterday

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