Reduction of Order Problem for Differential Equations Class

M87TJC
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Homework Statement
Find second solution for differential equation using reduction of order (see first image)
Relevant Equations
equation labeled (5) in first image in addition to equation 1
Problem statement:
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Second order linear differential equation in standard from
1633375439303.png

Reasoning:
1633376079966.jpeg
 

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I just looked back at the original problem, and I realized that I did not put the equation into standard form. If I divide the equation by 4 and repeat the same process I get the correct answer.
 
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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