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The topic of this problem is the complete response of RLC Circuits.

The problem is to find the second order differential equation expression for

the current i of t in the circuit show below.

So in our circuit we have a resistor, inductor, and

capacitor, and we have a current source i s of t.

We notice that the circuit is a single node pair circuit.

We have a ground node at the bottom of the circuit and

we have a node at the top of the circuit, where we've called node 1,

that'll have a voltage V sub 1 associated with it.

So if we're going to solve this problem using nodal analysis, or Kirchhoff's

current law, then we're going to sum the currents either into or out of node 1.

And we're choosing to sum the currents into node 1.

So if we do that we see from the right to left that we have the current

is(t) flowing into node 1, it's our first terminal expression.

We have a current which is flowing up through the resistor into node 1,

which is 0 volts minus V1 divided by R.

It's the second term in our expression.

We have a current i of t, which is flowing in the opposite direction,

it's flowing out of, so we have a minus i of t term.

And then we have also the current flowing

through the capacitor on the right hand side of our circuit.

And that's going to be c, dv, dt, and

so our c value and our voltage is 0 minus V1.

That's our V and our dvdt expression,

and the sum of those is equal to 0.

So if we use our well known equations which relate current and

voltage for the inductor, that is i of t is equal to 1 over L times

a integral of V1 of xdx plus the initial condition which we

are going to set equal to 0, so if we use this first expression.

And we use the second expression, which is a compliment of that first

expression for the inductor, that is the V is equal to L di, dt.

If we use those two expressions,

then we can rewrite our Kirchhoff's voltage law expression,

Look like this, we have L over R, di of t, dt.

So that's our V1 over R term.