# Math Help - Differential Equations in clinical medicine problem

1. ## Differential Equations in clinical medicine problem

The question is solve R(dVi/dt) + (1/C)Vi + Pex = Papp , 0 t ti

where

Vi (0) = 0

i, ex, app are subscripts

2. Originally Posted by targus2
The question is solve R(dVi/dt) + (1/C)Vi + Pex = Papp , 0 t ti

where

Vi (0) = 0

i, ex, app are subscripts
What is $R, C, P_{ex} \;\text{and}\; P_{app}$ ? Are they constants?

3. yes

4. So

$
\frac{d V_i}{dt} + \frac{V_i}{RC} = \frac{P_{app} - P_{ex}}{R}
$

The integrating factor is $e^{t/RC}$ so this gives

$
\frac{d}{dt} V_i e^{t/RC} = \frac{P_{app} - P_{ex}}{R} e^{t/RC}
$

Integrating gives

$V_i e^{t/RC} = C \left(P_{app} - P_{ex}\right) e^{t/RC} + V_0$

or

$
V_i = C\left(P_{app} - P_{ex}\right) + V_0 e^{-t/RC}.
$

Imposing the initial condition gives $V_0 = - C\left(P_{app} - P_{ex}\right)$

and thus the final solution

$V_i = C \left(P_{app} - P_{ex}\right) \left(1 - e^{t/RC} \right)$