# Linear ODE help

1. Dec 10, 2009

### titanae

1. The problem statement, all variables and given/known data
Eq 1: R(dV_i/dt) + (1/C)V_i + P_ex = P_app , 0 <= t <= t_i
Eq 2: R(dV_e/dt) + (1/C)V_e + P_ex = 0 , t_i <= t <= t_tot

A) Solve EQ 1 for V_i(t) with the initial condition V_i(0) = 0
B) Solve EQ 2 for V_e(t) with the initial condition V_e(t_i) = V_T
C) Using V_i(t_i) = V_T, show

P_ex = [((e^(t_i)/RC) - 1) * P_app] / ((e^(t_tot)/RC) - 1)

2. Relevant equations
V_i(0) = 0

V_e(t_i) = V_i(t_i) = V_T

V_e(tot) = 0

R, C, P_ex, P_app are constants

3. The attempt at a solution

A) V_i = C(P_app - P_ex)(1- (e^(t/RC)))

B) V_T = C(- P_ex)((e^(t/RC))-1)

C) ??

2. Dec 10, 2009

### ideasrule

I haven't checked A and B, but don't you just set V_i equal to V_T for C? Since V_i = C(P_app - P_ex)(1- (e^(t/RC))) and V_T = C(- P_ex)((e^(t/RC))-1), C(P_app - P_ex)(1- (e^(t/RC)))=V_T = C(- P_ex)((e^(t/RC))-1). Then solve for P_ex

3. Dec 10, 2009

### titanae

Yes i tried that and this is what i got

(P_app - P_ex)(1 - e^(t_i/RC)) = (-P_ex)(e^(t_tot/RC) - 1)

which when expanded out cancels P_ex

maybe another part is wrong?