Jtdylktuy

I am trying to find a solution for $C_a(t)$ in the below differential equation:

$$
V cdot frac {dC_a(t)} {dt}=R(t)-Q_v cdot C_a(t) \
$$
$
text{Where} \
qquad V text{ is the volume of the test chamber } [m^3] \
qquad C_a text{ is the concentration in test chamber } Big[frac{kg}{m^3} Big] \
qquad R text{ is the non-constant emission rate from test sample } Big[frac{kg}{m^2 cdot s} Big] \
qquad Q_v text{ is the constant volume flow rate of ventilation air passing through the test chamber } Big[frac{m^3}{s} Big] \
$
The non-constant emission rate is the "unknown function" mentioned in the question title. Stating that the function that governs emission rate is unknown is somewhat inaccurate as I have a vector containing values for emission for every examined time t.

Besides knowing values for $R$ at given times $t$, we know that $Ca(0) = 0$ and that $R(0) = 0$.

I am in the process of writing a small Matlab script, and therefore I am hoping to find a solution on a form that allows implementation and evaluation at given points in time (corresponding to the values in known vector $R$).

I have already given it my best, and I have arrived at the below expression (that I sincerely hope is correct). However, this expression considers $R(x)$ as a continuous function and will not be implemented easily.
$$
C_a(t) = exp(-Q_v t V^{-1}) int_{0}^{t}
exp(-Q_v x V^{-1}) R(x) V^{^-1} dx, quad for C_a(0)=0
$$
Can anybody help me find a simpler (explicit and fully analytical) solution for $C_a(t,R(t))$ that is easy to implement?

The solution with initial value $C_a(0) = C_0$ is
$$C_a(t) = exp(-Q_v t/V) left( C_0 + int_0^t exp(Q_v s/V) frac{R(s)}{V}; ds right) $$
You can approximate the integral with Riemann sums.

Turns out that Matlab does have a function that allows for solving ODEs with time dependent terms.

Thank you all very much for your time and help.