# Can I differentiate both side of a differential equation?

This is a trivial question, please note I'm not a professional in this environment, I'm just learning.

Let's suppose I've got this simple eqution:

$L\frac{d i(t)}{dt}=-\frac{1}{C}\int i(t) dt$

I suppose you guess what is, but is not so important: I want to solve it as a differential equation and so I need to remove the integral part. I know that the eq before can be written as:

$\frac{d^2 i(t)}{dt^2}+\frac{1}{LC}i(t) = 0$

It seems to me there is a differentiation of both side of the equation and a division of both sides by L. Is my guess correct ? If so, is it correct to say that differentiating both side of a differenctial equation yield to the same result? I suppose we need to state i(t) being differentiable twice, and maybe there is other restrictions...

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## migrated from physics.stackexchange.comDec 4 '12 at 23:51

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You are indeed correct, differentiating both sides of an equation yields a valid equation given that both sides are indeed differentiable.

Note, however, that the equation

$L\frac{d i(t)}{dt}=-\frac{1}{C}\int i(t) dt$

when rearranged and differentiated with respect to $t$ actually yields:

$\frac{d^2 i(t)}{dt^2}+\frac{1}{LC}i(t) = 0$

Hope this helps!

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yep, you are correct, I will accept as soon as I can: the addiotional dt was a typo. – Felice Pollano Dec 4 '12 at 20:33