Help with Second Order Circuits

[cshum00]

Homework Statement

I have a RLC circuit in series. I am trying to find the current i(t) passing through this RCL circuit. When solving the RCL circuit, it happened to be critically damped. And simplified i ended up with:
i(t) = A₂te^-20t
Where i need to find what is the cosntant A₂

Homework Equations

R = 100Ohm
L = 2.5H
C = 1mF
i(0) = 0A
v_C(0) = 24V
v_L = L di/dt
v_R = iR
di/dt = -20A₂te^-20t

The Attempt at a Solution

I apply KVL to the closed RCL loop and get:
v_C + v_L + v_R = 0
v_C(0) + L di(0)/dt + i(0)R = 0
24V + 2.5H di(0)/dt + 0 = 0
di(0)/dt = -24V/2.5H = -9.4A

But di/dt is also -20A₂te^-20t so
di(0)/dt = -9.4 = -20A₂(0)e^-20(0)
-9.4 = 0
which is fallacy

but according to the book:
A₂ = -9.4
so,
i(t) = -9.4te^-20t

Where did i go wrong?

 

[Mr.Green]

Dear Friend

di/dt = A2 * t * (-20 * e (-20t) ) + A2 * e(-20t)

The derivative of multiplied terms.

 

[cshum00]

Rgiht! How could i make such a basic mistake. The product rule! Thanks a bunch. T^T