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I am reading this doc about the equations for the step function for the RLC circuit, critically damped solution.

At page 16 we have these lines...

enter image description here

But there is a magic between these lines...

enter image description here

How the first line became the second one?

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    \$\begingroup\$ How not??? Are you familiar with the product rule? \$\endgroup\$ – jonk Aug 16 '18 at 22:31
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    \$\begingroup\$ The constant "B" is an integration constant. \$\endgroup\$ – user159625 Aug 16 '18 at 22:35
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    \$\begingroup\$ I think it may be time to switch authors for reading materials haha :) I feel like nearly all of your questions in the last two months have been about why this author decided to skip out important steps to the problem. \$\endgroup\$ – KingDuken Aug 16 '18 at 22:51
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    \$\begingroup\$ Sounds true. I remember my physics professor back in college was teaching us about some electromagnetics and he literally wrote an equation and he completely skipped some Stoke's Theorem and all of us were entirely confused. Well on the bright side, you're almost done reading this document haha. \$\endgroup\$ – KingDuken Aug 16 '18 at 22:55
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    \$\begingroup\$ It's far more instructive to research it yourself - 'independent learning'. Having everything laid on a plate does nothing to challenge and develop the intellect. \$\endgroup\$ – Chu Aug 17 '18 at 0:37
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This is simply the product rule in reverse:

$$\frac{d}{dt}[e^{\alpha t}v_{tr}] = e^{\alpha t}\frac{dv_{tr}}{dt} + v_{tr}\frac{d}{dt}[e^{\alpha t}] =e^{\alpha t}\frac{dv_{tr}}{dt} + \alpha e^{\alpha t} v_{tr}$$

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  • \$\begingroup\$ WOWWWWWWWWWWWWW! Brilliant. I was suspecting something on that direction but would never guess exactly that. THANKS! \$\endgroup\$ – SpaceDog Aug 16 '18 at 22:51

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