Please give me an idea for Reduction of order

  • Context: MHB 
  • Thread starter Thread starter Another1
  • Start date Start date
  • Tags Tags
    Idea Reduction
Click For Summary

Discussion Overview

The discussion revolves around the method of reduction of order in solving differential equations, specifically focusing on the transformation of a second-order differential equation into a first-order equation. Participants explore the implications of substituting a function into the equation and the resulting mathematical manipulations.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest starting with the substitution y = uv and inquire about the derivatives y' and y''.
  • There is a reiteration of the substitution leading to the expression y'' + Vy = uv'' + 2u'v' + u''v + Vuv = 0.
  • One participant notes that since u satisfies the equation u'' + Vu = 0, this leads to a simplification of the terms.
  • It is proposed that the order of the equation has been reduced from 2 to 1 through the manipulation of the resulting expressions.
  • Integration is discussed, leading to the expression for p = v' as Cu^{-2} and the subsequent integral for v(x).

Areas of Agreement / Disagreement

The discussion appears to be exploratory with no clear consensus reached. Participants build upon each other's contributions but do not explicitly agree on a single approach or solution.

Contextual Notes

Participants do not clarify all assumptions made during the derivations, and the discussion includes various steps that may depend on specific definitions or interpretations of the functions involved.

Another1
Messages
39
Reaction score
0
phpkJkbpx.png
 
Physics news on Phys.org
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.
 
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
 
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:

Similar threads

Undergrad Solving Differential Equation Using Reduction of Order
  • · Replies 2 ·
Replies
2
Views
4K
Are atomic conductivity and standard reduction potential correlated?
  • · Replies 5 ·
Replies
5
Views
2K
MHB Mike's question at Yahoo Answers regarding reduction of order
  • · Replies 1 ·
Replies
1
Views
2K
Please give me some ideas... (hackathon problem statement)
  • · Replies 7 ·
Replies
7
Views
1K
MHB Second order inhomogeneous differential equation
  • · Replies 1 ·
Replies
1
Views
2K
Insights Reduction of Order For Recursions
  • · Replies 4 ·
Replies
4
Views
3K
MHB Repeated roots, non homogeneous - second order, reduction of order method
  • · Replies 4 ·
Replies
4
Views
8K
MHB Method of Reduction of Order and Variation of Parameters
  • · Replies 2 ·
Replies
2
Views
5K
MHB 307.1.1 Use row reduction on the appropriate augmented
  • · Replies 5 ·
Replies
5
Views
1K
Compression ratio and volume reduction of CO2
  • · Replies 12 ·
Replies
12
Views
2K