Variation of Parameters Question

xlzhsteven
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Question is attached as Clipboard01.jpg

I have tried the use Variation of Parameters to solve this question, but I kept getting wrong answer.
This is What I get y=(2e^x)(Cos(e^x))+0.5(e^(-x))Cos(e^(-x))-2Sin(e^(-x))
This is the right answer: y=-Sin(e^(-x))-(e^x)Cos(e^(-x))

Procedure is shown in the second jpg file
Can someone help me with this problem?
Thank You
 

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Show how you got that answer and we will try to help.
 
I have attached my procedure in scan0002.jpg
 
Can anyone help me? I have math final tomorrow.
 
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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