# Homogeneous differential equation

## 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

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?

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gabbagabbahey
Homework Helper
Gold Member
With these types of problems, where you are given one solution you usually look for a second solution of the form $y_2(x)=u(x)y_1(x)$. So, try that substitution and see what you get.

With these types of problems, where you are given one solution you usually look for a second solution of the form $y_2(x)=u(x)y_1(x)$. 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?

gabbagabbahey
Homework Helper
Gold Member
y''=u''x+2u
You'll want to double check this

owho, yes yes, i get the answer. thank you very much.. ^^v