Question

Suppose f(x) is a function such that for some positive integer n, f has n linearly dependent derivatives. In other words, if f(x),f′(x),…,f(n−1)(x),f(n)(x) are all linearly dependent functions, then f(x) is expressible in terms of a,xk,eax,sin(ax),cos⁡(ax), and any combination of such functions, where a is a constant and k is a positive integer. Prove this statement

Answer 1

See explaination for the prove of the statement.

Explanation:

To establish this prove, lets refer back to what we already know.

We know that "If the set of reactions {d1,d2,d3,......dn} in a vector space V over a field f be linearly dependent, then atleast one of the vectors of the set can be expressed as a linear combination of the remaining others.

Please kindly go to attachment for a detailed step by step explaination of the prove.