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The timing of an intermittent windshield wiper is controlled by a variable resistor in a resistor-capacitor circuit. A 12 volt battery charges the capacitor. When the voltage across the capacitor reaches 10 volts, a switch closes to discharge the capacitor and energize the electric motor that moves the windshield wiper. Then the process repeats with the battery charging the capacitor again. Task 1 (5 points) The voltage across the capacitor follows the equation 1 dv 1 + v= vb dt RC RC Where R = 30,000 ohms (resistance), C = 0.0001 F (capacitance), v₂ = 12 V (battery voltage). Solve the ODE using the methods from 2.2 or 2.3 and write vas a function of time (show all your work). Also, plot your solution v(t) with the initial condition v = 0 (any plotting software is fine). Note that the units given are consistent, so there is no need to do any unit conversions to use the equation. Task 2 (5 points) Use the same C, vb, and initial condition values in Task 1. Let R be a constant, but its value still to be determined. Find the resistance that is required to have the windshield wiper move every 10 seconds? In other words, what resistance is required for the capacitor to reach 10 V in 10 seconds? (Show all your work). Also, plot v(t) for this case. Task 3 (5 points) Use the same C, vb, and initial condition values in Task 1. Let R be a constant, but its value still to be determined. Find the resistance that is required to have the windshield wiper move every 5 seconds? In other words, what resistance is required for the capacitor to reach 10 V in 5 seconds? (Show all your work). Also, plot v(t) for this case.