Proving Uniqueness of Complementary Solution for y(x)=y_c+y_p

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Homework Statement



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The Attempt at a Solution



I can show that the complementary solution y_c solves L[y]=0 and any initial conditions for a unique choice of the c_i's, using the standard "Wronskian and invertible matrix proof". I'm stuck on this part though, how can I prove it for y(x) = y_c + y_p?
 
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Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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