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
SUMMARY

The discussion focuses on the reduction of order technique in solving second-order differential equations, specifically using the substitution \( y = uv \) where \( u \) satisfies the equation \( u'' + V(x)u = 0 \). The transformation leads to a first-order equation by deriving \( y' \) and \( y'' \), resulting in the simplified form \( p' / p = -2u' / u \). This method effectively reduces the order of the equation from 2 to 1, allowing for easier integration and solution derivation.

PREREQUISITES
  • Understanding of second-order differential equations
  • Familiarity with the reduction of order technique
  • Knowledge of integration techniques
  • Basic concepts of differential calculus
NEXT STEPS
  • Study the method of reduction of order in detail
  • Learn about second-order linear differential equations
  • Explore integration techniques for solving differential equations
  • Investigate the implications of potential functions in differential equations
USEFUL FOR

Mathematicians, physics students, and engineers who are solving differential equations and seeking efficient methods for reducing their complexity.

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
Please give me some ideas... (hackathon problem statement)
  • · Replies 7 ·
Replies
7
Views
782
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 Second order inhomogeneous differential equation
  • · Replies 1 ·
Replies
1
Views
2K
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
Measuring ripple reduction in LTSpice
  • · Replies 22 ·
Replies
22
Views
7K