the auxiliary equation

Auxiliary equation is simply another name for the characteristic equation — the algebraic polynomial that a constant-coefficient linear ODE reduces to once you try an exponential solution. The two terms are completely interchangeable; different textbooks just happen to favour different words for the same object.

Starting from a y'' + b y' + c y = 0, you substitute the trial solution y = e^(r x). Since each derivative brings down a factor of r, the equation becomes (a r^2 + b r + c) e^(r x) = 0, and dividing by the never-zero exponential leaves the auxiliary equation a r^2 + b r + c = 0. It is 'auxiliary' in the sense of being a helper problem: you do not actually care about r for its own sake, but solving this little polynomial hands you the exponents that build the solution. The degree of the auxiliary equation matches the order of the ODE — second order gives a quadratic, n-th order gives a degree-n polynomial — and the nature of its roots (real distinct, repeated, or complex) selects which of the three solution forms you write down.

Why keep two names? 'Characteristic' emphasises that the polynomial captures the intrinsic character of the equation — its natural frequencies and decay rates. 'Auxiliary' emphasises its role as a stepping-stone you compute on the way to the answer. Recognising that they are synonyms saves confusion when you meet both in different sources; there is no subtle distinction hiding between them.