We start with:

`R(di)/(dt)+i/C=0`

Divide through by R:

`(di)/(dt)+(1/(RC))i=0`

We recognise this as a first order linear differential equation.

Identify P and Q:

`P=1/(RC)`

Q = 0

Find the integrating factor (our independent variable is t and the dependent variable is i):

`intP\ dt=int1/(RC)dt` `=1/(RC)t`

So

`IF=e^(t"/"RC`

Now for the right hand integral of the 1st order linear solution:

`intQe^(intPdt)dt=int0\ dt=K`

Applying the linear first order formula:

`ie^(t"/"RC)=K`

Since `i = V/R` when `t = 0`:

`K=V/R`

Substituting this back in:

`ie^(t"/"RC)=V/R`

Solving for i gives us the required expression:

`i=V/Re^(-t"/"RC)`

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