Stuck on the reduction of order step for solving this differential equation

[s3a]

Homework Statement

Find the general solution for the equation

(x - 1)y'' - xy' + y = sin(x), x > 1

Given that y_1(x) = e^x satisfies the associated homogeneous equation.

Homework Equations

y_2 = v_2(x) * y_1

The Attempt at a Solution

I read http://tutorial.math.lamar.edu/Classes/DE/ReductionofOrder.aspx and attempted to replicate its method several times and I am attaching my latest attempt. The website I linked to says "Note that upon simplifying the only terms remaining are those involving the derivatives of v. The term involving v drops out. If you’ve done all of your work correctly this should always happen." but I have a term involving v that did not drop out. Also, am I supposed to ignore sin(x) or not? Based on the way the question is phrased, I'd now say I should of ignored it (please tell me if I am correct in saying this) but it doesn't matter for what I am questioning.

By the way, this thread is just about the reduction of order part. (The next step is variation of parameters but I haven't gotten there yet.)

Any help would be greatly appreciated!
Thanks in advance!

 

[SammyS]

Homework Statement

Find the general solution for the equation

(x - 1)y'' - xy' + y = sin(x), x > 1

Given that y_1(x) = e^x satisfies the associated homogeneous equation.

Homework Equations

y_2 = v_2(x) * y_1

The Attempt at a Solution

I read http://tutorial.math.lamar.edu/Classes/DE/ReductionofOrder.aspx and attempted to replicate its method several times and I am attaching my latest attempt. The website I linked to says "Note that upon simplifying the only terms remaining are those involving the derivatives of v. The term involving v drops out. If you’ve done all of your work correctly this should always happen." but I have a term involving v that did not drop out. Also, am I supposed to ignore sin(x) or not? Based on the way the question is phrased, I'd now say I should of ignored it (please tell me if I am correct in saying this) but it doesn't matter for what I am questioning.

By the way, this thread is just about the reduction of order part. (The next step is variation of parameters but I haven't gotten there yet.)

Any help would be greatly appreciated!
Thanks in advance!

Check your algebra.

There is no term left involving v.

 

[s3a]

I found that the ve^x is supposed to be vxe^x such that they do cancel out (thanks) but now I'm stuck again. Could you please tell me what I am doing wrong now?

If I'm right so far, I don't see how what I did yields y = x.

 

[ehild]

You got v' correctly,

$$v'=C_1e^{-x}(x-1)$$.

Integrate, add second constant, multiply by e^x.

ehild

 

[s3a]

Sorry for the late reply but thanks :).

 

[ehild]

Better late than never

ehild

 

[s3a]

Lol ya. :D