MHB Please give me an idea for Reduction of order

[Another1]

 

[HOI]

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.

 

[Another1]

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

 

[HOI]

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},
$v(x)=\int^x \frac{d\chi}{\chi^2}$