The current I(t) in an LC series circuit is governed by the...

The current I(t) in an LC series circuit is governed by the initial value problem below. Determine the current as a function of time t. I''(t) + 25I(t) = g(t), I(0) = 3, I'(0) = 28, where g(t) = { 84 sin 2t, 0 ? t ? 2?; 0, 2? < t I(t) = 2 cos 5t + 5 sin 5t + 4 sin 2t + (8/5 sin 5t - 4 sin 2t)u(t - 2?)

Transcript

00:01 Hello student here in this question the given equation is i -dh -t plus 25 into i of t is equals to g of t where function g of t is given as 84 sine 2 when t lies between 0 to 2 pi and 0 when t is greater than 2 pi.

00:27 Now take lapless transformation on both side of this equation therefore we'll get lapless of i double dash t plus 25 i t is equals to laplace of g of t that is equals to laplace of i double h t plus 25 into laplace of i t is equal to integral from 0 to infinity e to the power minus st into g of t d t that is equals to s square into i of s minus s minus s into i h zero minus i -0 plus 25 into i of t is equals to integral from 0 to 2 pi it is to the power minus st into 84 sine of 2d d t plus integral from 2 pi to infinity it is to the power minus st into 0 into d t implies s square into i of s minus is given as 3 minus i dash at 0 is given as 28 plus 25 into i of s is equals to 84 over s squared plus 4 into e to the power minus s t into minus s into sine of 2d minus 2 goes 2d with lower limit 0 and upper limit 2 by implies s squared plus 25 into i of s minus 3 s plus 28 is equals to 168 over s squared plus 4 into 1 minus e t is to the power minus 2 pi s which implies i of s is equals to 168 divided by s square plus 4 into s square plus 25 into 1 minus e dash to the power minus 2 pi s plus 3 s plus 28 divided by s squared plus 25.

02:47 Using partial fraction will get i ofs is equals to 168 over 21 into 1 minus e to the power minus 2 pi s into 1 over s squared plus 4 minus 1 over s squared plus 25 plus 3 s over s squared plus 25 plus 28 over s squared plus 25.

03:15 Now take laplace inverse on both sides.

03:18 Therefore, we'll get laplace inverse of i ofs is equals to 84 into laplace invers of 1 over s squared plus 4 minus 84 into laplace inverso 1 over s squared plus 25 minus 84 into laplace inverso e -d -d -stile -bore minus 2 pi s over s over s...

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