Consider the given electric circuit consisting of a voltage source, resistor and capacitor in series.

In class we showed that Q satisfies the differential equation

R (dQ/dt) + (Q/C) = E(t)

Suppose R =1, C =2 and E(t) = sin t, so that the differential equation is

(dQ/dt) + (Q/2) = sin t

Furthermore, suppose that Q=1 when t = 0. Do the following.

a. Solve this initial value problem by hand. Show your work.

b. As t approaches infinity one of the terms in the solution goes to 0 and the solution approaches a steady state solution? What is this steady state solution? Explain how you arrived at your answer.

c. Use the DSolve function in Mathematica to solve the initial value problem. If necessary, show that the answer given by Mathematica is equivalent to the answer you obtained by hand. In parts c and d, turn in a printed copy of your input to and output from Mathematica.

d. Use Mathematica to make a plot that shows both the solution and the steady state solution for 0 ≤ t ≤ 15.