# Thread: 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 $\displaystyle R, C, P_{ex} \;\text{and}\; P_{app}$ ? Are they constants?

3. yes

4. 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

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