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

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

Â the current i of t in the circuit show below.

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

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

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

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

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

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

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

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

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

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

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

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

Â which is 0 volts minus V1 divided by R.

Â It's the second term in our expression.

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

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

Â And then we have also the current flowing

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

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

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

Â That's our V and our dvdt expression,

Â and the sum of those is equal to 0.

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

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

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

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

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

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

Â If we use those two expressions,

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

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

Â So that's our V1 over R term.

Â