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Figure I have problem with solving this rlc circuit, which we should detemine i for t>0. I have written kvl law and got this equations:`

0.04i'' + 1.21 i' + 4i = -50 which has this characterstic equation: 0.04s^2 + 30.25s + 100 = 0

s = -3.7, s= -26.4.

Are these equations correct? Unfortunately the solution only has the final answer which is different than mine. I'm currently self-studying so i couldn't ask anywhere else. would you please solve this circuit and determine i?

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Well, assuming that all the initial conditions are equal to \$0\$. We can start the analysis.

We know that:

$$\frac{50}{\text{s}}=\text{i}_\text{in}\left(\text{s}\right)\cdot\left(6+\frac{\text{s}}{4}+\frac{4\cdot\frac{25}{\text{s}}}{4+\frac{25}{\text{s}}}\right)\space\Longleftrightarrow\space\text{i}_\text{in}\left(\text{s}\right)=\frac{50}{\text{s}}\cdot\frac{1}{6+\frac{\text{s}}{4}+\frac{100}{4\text{s}+25}}\tag1$$

And we get:

$$\text{i}\left(\text{s}\right)=-\frac{25}{\text{s}}\cdot\frac{1}{\frac{25}{\text{s}}+4}\cdot\text{i}_\text{in}\left(\text{s}\right)=-\frac{25}{25+4\text{s}}\cdot\frac{50}{\text{s}}\cdot\frac{1}{6+\frac{\text{s}}{4}+\frac{100}{4\text{s}+25}}\tag2$$

Taking the inverse Laplace transform will to lead to this.

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