1st order linear ODE example: RC circuit. Solve for I(t) in response to a step i

Question

1st order linear ODE example: RC circuit. Solve for I(t) in response to a step input of magnitude V0. At time t = 0, the current will be nonzero (say, I0). However, you will need to solve for I0 so that your final expression of is I(t) in terms of V0 and the system parameters. There should not be any undefined constant terms in your answer. Hint: Your general solution from Prob. 1a should simplify quickly once you consider the step input and its rate of change. For the purposes of this simple homework problem, don't worry about the Dirac Delta function.

Explanation / Answer

let at a time t , charge on capacitor is q , icurrent is flowing in the circuit.

apply KVL,

Vin = iR +q/c
Vin = Rdq/dt +q/c      , becuase i =dq/dt

after integrating , q = cVin + Ae-t/Rc       , where A = integration constant

                              i(t) =dq/dt   = (A/Rc)e-t/Rc ...........(1)

               given   at t=0 , current is Io , then Io =A/Rc or   A =RcIo

now from (1)   i(t) = Ioe-t/Rc

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