Please give me an idea for Reduction of order

In summary, we have tried using the fact that u satisfies the equation u''+ V(x)u= 0 to find the solution for y''+ Vy= 0. By integrating both sides and taking the exponential, we have found the solution to be v(x)=\int^x \frac{d\chi}{\chi^2}.
  • #1
Another1
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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.
 
  • #3
Country Boy said:
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
 
  • #4
Another said:
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},
$v(x)=\int^x \frac{d\chi}{\chi^2}$
 
Last edited:

1. What is reduction of order in scientific terms?

Reduction of order is a mathematical method used to simplify a system of differential equations by reducing the order of the highest derivative present in the equations.

2. Why is reduction of order important in scientific research?

Reduction of order allows scientists to simplify complex systems and make them easier to analyze and understand. This can lead to more accurate predictions and insights into the behavior of the system.

3. How is reduction of order different from other mathematical methods?

Reduction of order specifically focuses on simplifying systems of differential equations, while other methods may be more general and applicable to a wider range of mathematical problems.

4. Can reduction of order be applied to any system of differential equations?

No, reduction of order can only be applied to systems of differential equations that meet certain criteria, such as being linear and homogeneous.

5. Are there any limitations to using reduction of order?

Reduction of order may not always provide the most accurate results, as it involves simplifying the system and may not take into account all factors and variables. It is important to carefully consider the limitations and assumptions of this method before applying it to a scientific problem.

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