1. The problem statement, all variables and given/known data
A resistor with resistance R and a capacitor with capacitance C are connected in series to a direct current battery ε.
find the current and charge on the circuit as function of time.

it looks more like a review of differential equations, so I’m not really sure if I should post here or in the calculus forum….feel free to move it if you think it’s better.

2. Relevant equations

potential at resistor: $V_r = Ri$

potential at capacitor:$V_c = \frac{q}{C}$

3. The attempt at a solution

Applying one of the Kirchhoff’s law:
$\epsilon – Ri – \frac{q}{C} = 0$
$\epsilon = Ri +\frac{q}{C}$
since $i = \frac{dq}{dt}$ we can rewrite the equation as

$\frac{\epsilon}{R} = \frac{dq}{dt} + \frac{q}{RC}$ and solve with an integration factor $e^{\frac{t}{RC}}$
so we have:
$\frac{\epsilon}{R} e^{\frac{t}{RC}} = \frac{d}{dt}(q e^{\frac{t}{RC}})$
and then:

$\int \frac{\epsilon}{R} e^{\frac{t}{RC}} dt = q e^{\frac{t}{RC}}$

here I’m kinda stuck:
because the problem did not give any initial conditions, should I just solve an indefinite integral or integrate from zero to an arbitrary t??

also: $q e^{\frac{t}{RC}}$ comes from integrating $\int_{a}^{b} \frac{d}{dt}(q e^{\frac{t}{RC}})dt$ and applying the fundamental theorem of calculus.
However, don’t I need the values a and b to properly use it?

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