Homogeneous differential equation

[annoymage]

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=e^rx
and also i can't use substitution y=x^r

how to do this?

 

[gabbagabbahey]

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.

 

[annoymage]

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]

y''=u''x+2u

You'll want to double check this

 

[annoymage]

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