Mathematically modelling RC circuit with a linear input
Unfortunately, I have found no resource that discusses how to
  mathematically model an RC circuit, were one to provide a linearly
  increasing voltage source as an input.


This answer is all about converting the circuit to a transfer function in the frequency domain then multiplying that T.F. with the Laplace transform of the input to get the frequency domain equivalent of the output. Finally, a reverse Laplace operation is performed to obtain the time domain formula for the output.

The Laplace transform of a low pass RC filter is: -

$$\dfrac{1}{1+sRC}$$

This is the frequency domain transfer function so, if you multiply this by the frequency domain equivalent of a ramp (\$\dfrac{1}{s^2}\$) you get the frequency domain output: -

$$\dfrac{1}{s^2(1+sRC)}$$

Using a reverse laplace transfer table this has a time domain output of: -

$$t + RC\cdot e^{(\frac{-t}{RC})}-RC$$

See item 32 on the table or, if the formula didn't have an obvious table entry you can use an inverse laplace calculator that solves it numerically like this one.

The calculator allows you to build the formula and enter a numerical value for RC. I used an RC value 7 in the above example so I could see how that number propagated to the final answer. The last hurdle is substituting that propagated value of 7 with RC. In other words it's a numerical solver but nevertheless a very useful tool to have to hand: -