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## Homework Statement

Suppose that we have the following linear ODE:

[tex]y^{(n)} + a_{n-1}y^{(n-1)} + a_{n+2}y^{(n-2)} + ... + a_2y'' + a_1y' + a_0y = 0[/tex]

Prove that if we have (n-1) linearly independent {y

_{1},...,y

_{n-1}} solutions then we can find y

_{n}.

## The Attempt at a Solution

well, this is my idea:

Imagine that C(a,b) is the linear space of all the functions that are continuous in the interval [a,b]. Let's define an inner product on this space as the following:

<f,g> = ∫

_{a}

^{b}f(x)g(x)dx

Now, Imagine I have n-1 linearly independent functions y

_{1},...,y

_{n-1}then the problem is transformed into finding a function y

_{n}such that y

_{n}is perpendicular to all the other y

_{i}'s (i=1,...,n-1).

well, it means I should show that there exists a function that for any y

_{i}I have:

∫

_{a}

^{b}y

_{n}y

_{i}dx

Here is where I'm stuck. What should I do?

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