I need an anwser for Extra credit problem 9) Let vin(t) be applied to the input

Question

I need an anwser for Extra credit problem

9) Let vin(t) be applied to the input resistor of the ideal op circuit shown. Assume that the capacitor is initially uncharged. What is the output of the op amp, and what mathematical operation is the circuit performing? (6) EXTRA CREDIT PROBLEM: Design an ideal op amp circuit (with component values) that will output a voltage v(t) = cos(10t). Note that the differential equation, d^2v/dv^ 2 +100v = 0 has a solution v(t) = cos(10t) if we impose the initial conditions y(0) = 1 and dv(0)/dt = 0. Refer to my first lecture on Chapter 5.(25)

Explanation / Answer

calculating in s domain(laplace domain):

capacitor will be 1/(s*C)

then as it is an ideal op-amp, using virtual ground concept,

voltage at -ve node=0

then using KCL at this node:

(Vin-0)/R=(0-Vout)/(1/(sC))

Vin/R=-s*C*Vout

Vout=-Vin/(s*R*C)

using inverse laplace transform:

Vout=-(1/RC)*integration of Vin(t)*dt

so here we are performing the integration action

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