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.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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