Prove: The contribution to the general solution from a set of q equal roots with value m = b is
y = c1ebx + c2xebx + ........ + cqxq -1ebx
Proof. We want to find q linearly independent solutions to the equation corresponding to the q equal roots. If the auxiliary equation f(m) = 0 has q equal roots m1 = m2 = ........ = mq = b, then the operator f(D) must have a factor (D- b)q. We want to find q linearly independent y’s for which
(D- b)q y = 0
We employ the following theorem.
Theorem. (D - m)n xkemx. = 0 for k = 0, 1, 2, .... , (n - 1) i.e. for all k < n
(see section Differential Operators, Some theorems involving emx and xkemx, Theorem 4 for the theorem and its proof)
Replacing n by q and m by b in the theorem, we get
(D - m)q xkebx. = 0 for k = 0, 1, 2, .... , (q - 1) i.e. for all k < n
which means there are q solutions
c1ebx, c2xebx, ........ , cqxq -1ebx
to the equation. These functions are linearly independent because, aside from the common factor ebx, they contain only the respective powers x0, x1, x2, .... , xq -1 and the set x0, x1, x2, .... , xq -1 is a linearly independent set of functions.
1. Earl D. Rainville. Elementary Differential Equations