# Laplace Transform for ODE - RC Circuit

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Guilherme Lima on 24 Mar 2021
Edited: darova on 27 Mar 2021
Hello guys,
I am trying to make a laplace transform from a ODE for a simple RC Circuit.
But When try to use the function subs() in order to replace my laplace transfromed variable >>> laplace(q(t), t, s) for Q(s) I don't see any change at all.
I would like to undestand why and How I can fix this.
%For the capacitor
syms q(t) s
%Charge (C)
%initial condition in the beginning
cond_q = q(0) == 0;
%ODE for capacitor in series with resistor
ode_capacitor = diff(q, t) + P_c*q == Q_c
%Solving ODE
q = dsolve(ode_capacitor, cond_q);
% simplify(q)
% % % Calculate the derivative of 'Charge' - the current(A)
i_c = diff(q)
limit(i_c, t, 0)
limit(i_c, t, inf)
%
laplace(i_c)
a = laplace(ode_capacitor, t, s)
syms Q
a = subs(a, laplace(q, t, s), Q)

Paul on 25 Mar 2021
I think that last subs command is getting confused because q is defined as the solution of the ode. I'm not sure why that's a problem (or if I'm even correct about that), but you can get around this by using a different variable name as the solution of dsolve:
syms q(t) Q(s) P_c Q_c
cond_q = q(0) == 0;
ode_capacitor = diff(q(t),t) + P_c*q == Q_c;
qsol(t) = dsolve(ode_capacitor,cond_q);
i_c(t) = diff(qsol(t),t);
I_c(s) = laplace(i_c(t));
% Laplace transform of the ode
E(s) = laplace(ode_capacitor);
E(s) = subs(E(s),laplace(q(t),t,s),Q(s));
E(s) = isolate(E(s),Q(s));
Q(s) = rhs(E(s));
Q(s) = subs(Q(s),q(0),0);
% compare Q(s) with laplace(qsol(t))
[Q(s) simplify(laplace(qsol(t)))]
% verify relationship between charge and current
[s*Q(s) I_c(s)]
ans =
[ Q_c/(s*(P_c + s)), Q_c/(s*(P_c + s))]
ans =
[ Q_c/(P_c + s), Q_c/(P_c + s)]
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darova on 26 Mar 2021