One equation can be easily obtained using the limit trick. We multiply the general decomposition by y and pass to infinity.

So we have one constant and we need five equations to get the others. Besides the multiplication method we can also try the plug-in method.

As usual, getting these equations was relatively simple, but we paid for it with large coefficients.

How would we solve these five equations effectively if we did not feel like row elimination? A good idea is to use symmetry for pair of numbers. The last two equations have the same coefficients, just ocassionally with different signs. If we subtract them, we get an equation featuring just C and E. Similarly by subtracting the two equations before them we get another equation with just C and E. We have two equations with two unknowns, which is a simple problem to solve. The constants that we obtained can be put for instance into the first, second and the fourth equation and we get three equations with three unknowns, which is a reasonable problem.