Find particular solution for both UESTION #2 (10 MARKS EACH) 1. The differential

Question

Find particular solution for both

UESTION #2 (10 MARKS EACH) 1. The differential equation that controls the circuit with inductance L, current i and resistance R is di dt L -+ Ri = E cos(at) where E and are constants. If i(0) = 0, find the particular solution 2. Consider an electric circuit containing a capacitor, resistor and battery. The charge q(t) on the capacitor satisfies the equation dq q dt C where R is resistance, V is constant voltage and C is capacitance. If q(0) = 0, obtain the particular solution.

Explanation / Answer

1) j=RiE to turn the equation into

LRdjdt=jj=keRLt

So the solution is

i=j+ER=CeRLt+ER

for arbitary constant C (which is to related to the arbitrary constant k by C=k/R).

2)

RdQdt+1CQ=0 Dividing through by R gives : dQdt+1RCQ=0 Now as posted by other people, 1RC is a constant just like 2,3 and any integer you can think of. So when finding the integrating factor, it is necessary to remember that we are integrating with respect to t, not R or C : IntegratingFactor=e1RCdt=e1RCt Now we multiply both sides of the differential equation by the integrating factor: e1RCtdQdt+e1RCt1RCQ=0 Now we integrate both sides with the respect to t : e1RCtQ=0dt=c where c is the constant of integration. Finally, we get the equation in terms of Q by dividing both sides by the integrating factor: Q=c/e^t/RC

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