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Please give me an idea for Reduction of order

Another

Member
Feb 11, 2018
39
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Country Boy

Well-known member
MHB Math Helper
Jan 30, 2018
464
What have you tried? If y= uv then what is y'? What is y''? What do you get when you put those into the differential equation? And then use the fact that u itself satisfies the equation, that u''+ V(x)u= 0.
 

Another

Member
Feb 11, 2018
39
What have you tried? If y= uv then what is y'? What is y''? What do you get when you put those into the differential equation? And then use the fact that u itself satisfies the equation, that u''+ V(x)u= 0.
y'' = uv'' +2u'v'+ u''v

so

y''+ Vy = uv'' +2u'v'+ u''v + Vuv = 0
 

Country Boy

Well-known member
MHB Math Helper
Jan 30, 2018
464
y'' = uv'' +2u'v'+ u''v

so

y''+ Vy = uv'' +2u'v'+ u''v + Vuv = 0
Okay, and since u satisfies u''+ Vu= 0, that is
uv''+ 2u'v'+ v(u''+ Vu)= uv''+ 2u'v'= 0.

uv''= -2u'v'.

Let p= v'. Then up'= -2u'p so that p'/p= -2u'/u.
The order of the equation has been "reduced" from 2 to 1.

Integrating both sides ln(p)= -2ln(u)+ c= ln(u^{-2})+ c.

Taking the exponential of both sides, p= v'= Cu^{-2},
$v(x)=\int^x \frac{d\chi}{\chi^2}$
 
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