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...
But there is a magic between these lines...
How the first line became the second one?
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3\$\begingroup\$ How not??? Are you familiar with the product rule? \$\endgroup\$– jonkCommented Aug 16, 2018 at 22:31
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1\$\begingroup\$ The constant "B" is an integration constant. \$\endgroup\$– user159625Commented Aug 16, 2018 at 22:35
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2\$\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\$– user103380Commented Aug 16, 2018 at 22:51
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1\$\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\$– user103380Commented Aug 16, 2018 at 22:55
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1\$\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\$– ChuCommented Aug 17, 2018 at 0:37
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1 Answer
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This is simply the product rule in reverse:
$$\frac{d}{dt}[e^{\alpha t}v_{tr}] = e^{\alpha t}\frac{dv_{tr}}{dt} + v_{tr}\frac{d}{dt}[e^{\alpha t}] =e^{\alpha t}\frac{dv_{tr}}{dt} + \alpha e^{\alpha t} v_{tr}$$
answered Aug 16, 2018 at 22:31
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\$\begingroup\$ WOWWWWWWWWWWWWW! Brilliant. I was suspecting something on that direction but would never guess exactly that. THANKS! \$\endgroup\$– DuckCommented Aug 16, 2018 at 22:51