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Homogeneous differential equation

  • Thread starter annoymage
  • Start date
  • #1
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Homework Statement



(1-xcotx)y''-xy'+y=0

y1(x)=x is a solution

find the second solution, y2(x), y1 and y2 are linear independent

Homework Equations



N/A

The Attempt at a Solution



i only know how to find it by auxiliary equation by substitute y=erx
and also i cant use substitution y=xr

how to do this?
 
Last edited:

Answers and Replies

  • #2
gabbagabbahey
Homework Helper
Gold Member
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With these types of problems, where you are given one solution you usually look for a second solution of the form [itex]y_2(x)=u(x)y_1(x)[/itex]. So, try that substitution and see what you get.
 
  • #3
362
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With these types of problems, where you are given one solution you usually look for a second solution of the form [itex]y_2(x)=u(x)y_1(x)[/itex]. So, try that substitution and see what you get.
y=ux

y'=u'x+u

y''=u''x+2u

substitute and get

-(x2u''+2xu)cot(x)-x2u'+xu''+2u=0

like this right?

but it seems to become more complicated is it? or should i continue?
 
  • #5
362
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owho, yes yes, i get the answer. thank you very much.. ^^v
 

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