Finding y_2(x) using Reduction of Order

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Homework Statement


Find y_2(x) given that y_1(x) = e^x for (x – 1)y'' – xy' + y = sin x.

Homework Equations


Reduction of order method.

The Attempt at a Solution


My attempt is attached as MyWork.jpg. Is what I did so far 100% correct? Assuming it is, what do I do now?

Any help in figuring this out would be greatly appreciated!
Thanks in advance!
 

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I figured it out.
 

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