Assume .
Evaluating at ,
Differentiating once,
Evaluating at ,
etc.
Thus, .
So, are linearly independent.
Is this right?
Prove that an nth-order homogenous linear differential equation satisfying the hypotheses of the Existence and Uniqueness Theorem for Linear Equations has n linearly independent solutions .
(Suggestion: Let be the unique solution such that
and if .)