The question is solve R(dVi/dt) + (1/C)Vi + Pex = Papp , 0 ≤ t ≤ ti
where
Vi (0) = 0
i, ex, app are subscripts
So
$\displaystyle
\frac{d V_i}{dt} + \frac{V_i}{RC} = \frac{P_{app} - P_{ex}}{R}
$
The integrating factor is $\displaystyle e^{t/RC}$ so this gives
$\displaystyle
\frac{d}{dt} V_i e^{t/RC} = \frac{P_{app} - P_{ex}}{R} e^{t/RC}
$
Integrating gives
$\displaystyle V_i e^{t/RC} = C \left(P_{app} - P_{ex}\right) e^{t/RC} + V_0$
or
$\displaystyle
V_i = C\left(P_{app} - P_{ex}\right) + V_0 e^{-t/RC}.
$
Imposing the initial condition gives $\displaystyle V_0 = - C\left(P_{app} - P_{ex}\right)$
and thus the final solution
$\displaystyle V_i = C \left(P_{app} - P_{ex}\right) \left(1 - e^{t/RC} \right)$