Prove: The contribution to the general solution from a set of q equal roots with value m = b is

            y = c₁e^bx + c₂xe^bx + ........ + c_qx^{q -1}e^bx

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 = ........ = m_q = 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 x^ke^mx. = 0  for k = 0, 1, 2, .... , (n - 1) i.e. for all k < n

(see section Differential Operators, Some theorems involving e^mx and x^ke^mx, Theorem 4 for the theorem and its proof)

Replacing n by q and m by b in the theorem, we get

            (D - m)^q x^ke^bx. = 0   for k = 0, 1, 2, .... , (q - 1) i.e. for all k < n

which means there are q solutions

            c₁e^bx, c₂xe^bx, ........ , c_qx^{q -1}e^bx

to the equation. These functions are linearly independent because, aside from the common factor e^bx, they contain only the respective powers x⁰, x¹, x², .... , x^{q -1} and the set x⁰, x¹, x², .... , x^{q -1} is a linearly independent set of functions.

References

1. Earl D. Rainville. Elementary Differential Equations