To solve the problems in this section, use the method of convolution from Section 6.4 of the book. Use also the formulas that were derived in Section 6.4 from the concept of convolution.
IVP = Initial Value Problem
1.
Consider an electric circuit as described in Section 3.1. Our electric circuit has a resistance \(R\) of 3 M\(\Omega\text{,}\) an electric voltage source \(E\) of 3 Volt, and a capacity \(C\) of 1 \(\mu\)Farads. The IVP that describes the changes in the voltage drop \(E_C(t)\) across the capacitor of the circuit is:
Solve this IVP using the formulas derived in Section 6.4 from the concept of convolution.
How is convolution different than the previous methods you have learned?
How is convolution an advantage to previous methods you have learned?
How is convolution a disadvantage to previous methods you have learned?
2.
Consider another electric circuit as described in Section 4.1. This time, our electric circuit has a resistance \(R\) of 0 \(\Omega\text{,}\) a capacity \(C\) of \(\frac{1}{4}\)Farads, an inductance \(L\) of 1 Henry, and an electric voltage source \(E\) of \(2\) Volts. There is also a switch that is turned on at time \(t=\pi\) seconds and is turned off at time \(t=2\pi\) seconds. That results in the forcing function being\(2(u(t-\pi)-u(t-2\pi))\) Volts. So the IVP that describes the changes in the electric charge \(Q(t)\) on the capacitor of the circuit is:
Graph a phase diagram of the response with \(Q(t)\) on the horizontal axis and \(Q'(t)\) on the vertical axis.
What have you learned about the workings of our electric circuit from a, b, and c?
3.
Consider a pendulum that is hanging at rest. After 1 second, you hit it lightly with a hammer. After another second, you hit it again with the exact same force in the same direction. The change of the angle \(\theta(t)\) from the center point can be described as:
