finding current and charge in a RC circuit.

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: [itex] V_r = Ri [/itex]

potential at capacitor:[itex] V_c = \frac{q}{C} [/itex]

3. The attempt at a solution

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

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

[itex] \int \frac{\epsilon}{R} e^{\frac{t}{RC}} dt = q e^{\frac{t}{RC}} [/itex]

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: [itex] q e^{\frac{t}{RC}} [/itex] comes from integrating [itex] \int_{a}^{b} \frac{d}{dt}(q e^{\frac{t}{RC}})dt [/itex] 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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