Here is an interesting problem recently shared by Neeraj.
As far as I can tell, there are infinitely many solutions, because when you write a system of four linear equations, the system matrix always ends up being singular. Let be the capacitor voltages. To get a solution, declare the value of and solve a system of three linear equations to get the other three voltages. The system of three linear equations is non-singular.
The following equations describe the system (being mindful of the capacitor polarities):
Since we are declaring the value of , move it over to the other side:
Recall that is the battery voltage. For , . For , . These are just two solutions out of infinitely many possibilities.
One other thing to note. The following relation also holds:
So if is chosen as a positive value, will be negative. If one wishes to impose an additional constraint that none of the capacitor voltages can be negative, it follows that and are both zero.