Reduction of Order: Integration Explained

  • Thread starter badman
  • Start date
  • Tags
    Reduction
In summary, the conversation discusses a problem involving integration and partial fractions. The solution is obtained by setting up a system of equations and using substitution to solve for the integral. The final formula is given as w=c(x^2+1)/x^2.
  • #1
badman
57
0
hello everyone, I am stuck on this problem and i have a hard time figuring out how they went from here dw/w=-(2dx/x(x^2+1)) to here w=c(x^2+1)/x^2. i know they integrated, but can anyone show me the details of the integration along with an explanation?
 
Physics news on Phys.org
  • #2
partial fractions
dw/w = -2 dx/ (x*(x^2+1))
= (-2/x + 2x/(x^2+1) )dx
 
  • #3
qbert used "partial fractions". The fraction can be written
[tex]\frac{-2}{x(x^2+1)}= \frac{A}{x}+ \frac{Bx+ C}{x^2+ 1}[/itex]
Multiplying on both sides by that denominator,
[tex]-2= A(x^2+ 1)+ (Bx+ C)(x)[/tex]
Let x= 0 and that becomes -2= A.
Let x= 1 and we have -2= (-2)(2)+ B+ C or B+ C= 2.
Let x= -1 and we have -2= (-2)(2)+ B- C or B- C= 2.
Adding, 2B= 4 or B= 2 and C= 0.
That gives the formula qbert wrote.
Of course, the integral of -2/x is -2 ln(x) and the the integral of [itex]\frac{2x}{x^2+1}[/itex] can be done by the substitution u= x2+ 1.
 
Last edited by a moderator:

1. What is reduction of order?

Reduction of order is a mathematical technique used to simplify the process of solving higher order differential equations. It involves converting a higher order equation into a first order equation, making it easier to integrate and find the solution.

2. How does reduction of order work?

Reduction of order works by using a substitution method to convert a higher order differential equation into a first order equation. This involves introducing a new variable and using the chain rule to rewrite the equation in terms of the new variable. The resulting first order equation can then be integrated to find the solution.

3. When is reduction of order used?

Reduction of order is typically used when solving homogeneous second order linear differential equations with constant coefficients. It can also be used for higher order equations, but the process becomes more complex.

4. What are the benefits of using reduction of order?

The main benefit of using reduction of order is that it simplifies the process of solving higher order differential equations. It also allows for the use of techniques such as separation of variables and integrating factors, which may not be possible with the original higher order equation.

5. Are there any limitations to reduction of order?

Reduction of order may not always be possible or practical for solving higher order differential equations. It also does not work for non-linear equations or those with non-constant coefficients. Additionally, the resulting first order equation may still be difficult to integrate, requiring additional techniques.

Similar threads

I Solving Differential Equation Using Reduction of Order
    • Differential Equations
Replies
2
Views
983
I Solving 2D Heat Equation w/ FEM & Galerkin Method
    • Differential Equations
Replies
7
Views
2K
I Is it possible to solve such a differential equation?
    • Differential Equations
Replies
20
Views
2K
How should I find the nontrivial stationary paths?
    • Calculus and Beyond Homework Help
Replies
10
Views
345
MHB B.2.1.4 trig w/ integrating factor
    • Differential Equations
Replies
1
Views
860
MHB Solving a Riccati Differential Equation
    • Differential Equations
Replies
1
Views
2K
A Numerical Solution to Random Linear Non-Homogeneous ODE
    • Differential Equations
Replies
1
Views
984
POTW Contour Integral Representation of a Function
    • Math POTW for Graduate Students
Replies
4
Views
1K
I Real ODE yields real solution through complex numbers
    • Differential Equations
Replies
3
Views
2K
I Assuming boundary conditions when integrating by parts
    • Differential Equations
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top