Hello, I have spent a lot of time trying to understand why I can't do this exercise by two different ways and find the same result.

I have to find Vc1(t) t ≥ 0 by two ways

1. first find the differential equation and then apply laplace transformation
2. second analyzing the circuit directly in the frequency domain

Solution:
1. find the differential equation and then apply laplace transformation

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2. analyzing the circuit directly in the frequency domain

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as you can see both answers are different and this this makes no sense..... I really don't know why HELP!!!

 

Your first solution is wrong.
First, for the integral, you should take limits into account. By the way, why don't you Laplace and solve for I(s) in method 1?

 

I put the initial conditions after transform the differential equation

 

I put the initial conditions after transform the differential equation

Here is my result when solve for V1(t):

 

Great!!, finally I undestood my mistake thank you