I am trying to figure out what exactly happens to the capacitor with 50 V when we connect it to a power source with 25 V. It probably discharges but I am wondering at what rate it stabilizes and at how much T it stabilizes. Here is a picture of the problem.
You show the power supply as a battery. Most batteries, both primary and secondary, can absorb current in the reverse direction. The capacitor will discharge into the battery, the rate depending on the internal resistance of the battery plus the 10K resistor. With secondary cells it will just charge the battery a bit.
If your source is actually a bench power supply then the result depends upon the design of the supply. There are three possibilities I can think of.
You will blow the output circuit of the power supply! An unlikely scenario but depending on the voltage ratings of the components and their ability to absorb reverse current.
It will be able to safely absorb current and act very similar to a battery, draining the capacitor till it is at 25V.
It is unable to absorb reverse current (like a reverse biased diode) and it will just sit there with the terminals of the power supply at 50V.
But assuming that the power source is able to absorb current and has a negligible internal resistance compared to 10K then you would see the usual exponential decay of the capacitor voltage with a time constant of CR, just superimposed on a 25V level.
