(Solved): Differential Equation A series circuit contains a resistor and a capacitor as shown in Figure 1. Sup ...

A series circuit contains a resistor and a capacitor as shown in Figure 1. Suppose the capacitor in the circuit initially has zero charge and that there is no initial current. When the switch is at position B, the $10\mathrm{Volt}$ power supply energized the capacitor through the $250?$ resistor, until fully charged, thus producing an output voltage $E_0$. At time $t=2$ seconds, the switch is thrown from position B to A, held there for 1 second, then switch back to B. Figure 1 (a) Write the differential equation for the charge $q(t)$ of the circuit. (b) Given the forcing function is zero until $t=2$, then has value 10 volts until $t=3$, then is zero again, express $E(t)$ in terms of unit step function and sketch the function $E(t)$. (c) Find the charge on the capacitor using part (a) and (b). Use the Laplace transform method. (d) What is the output voltage $E(t)$ on the capacitor?