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

y = c_{1}e^{bx} + c_{2}xe^{bx} + ........ + c_{q}x^{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 m_{1} = m_{2} = ........ = 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^{k}e^{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^{k}e^{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^{k}e^{bx.} = 0 for k = 0, 1, 2, .... , (q - 1) i.e. for all k < n

which means there are q solutions

c_{1}e^{bx}, c_{2}xe^{bx}, ........ , c_{q}x^{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 x^{0}, x^{1}, x^{2}, .... , x^{q -1} and the set x^{0}, x^{1}, x^{2}, .... , x^{q -1} is a
linearly independent set of functions.

References

1. Earl D. Rainville. Elementary Differential Equations

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