Efficient Solutions for Solving First Order ODE with Constant A

  • Context: Graduate 
  • Thread starter Thread starter BobbyBear
  • Start date Start date
  • Tags Tags
    Ode
Click For Summary
SUMMARY

The discussion focuses on solving the first order ordinary differential equation (ODE) given by \(\frac{dz}{dx} = \frac{z(x^4z^2-A^2)}{x(A^2-x^2z^4)}\), where A is a constant. The ODE is identified as neither exact nor homogeneous, presenting a challenge for standard solution techniques. An integrating factor can be utilized to transform the ODE into an exact form, leading to the general solution expressed implicitly as \(x^2 + \frac{z^2 + A^2}{(xz)^2} = C\), where C is a constant. A clever method is suggested to solve for z from this implicit equation.

PREREQUISITES
  • Understanding of first order ordinary differential equations (ODEs)
  • Familiarity with integrating factors in differential equations
  • Knowledge of implicit functions and their solutions
  • Basic algebraic manipulation skills
NEXT STEPS
  • Research methods for finding integrating factors for non-exact ODEs
  • Learn about implicit function theorem and its applications
  • Explore advanced techniques for solving first order ODEs
  • Study examples of ODEs transformed into exact equations
USEFUL FOR

Mathematicians, engineering students, and anyone involved in solving differential equations, particularly those dealing with first order ODEs and integrating factors.

BobbyBear
Messages
162
Reaction score
1
Hello,
does anyone have any ideas on how to solve this first order ode?

[tex]\frac{dz}{dx} = \frac{z(x^4z^2-A^2)}{x(A^2-x^2z^4)}[/tex]

where A is a constant.

It's neither exact, nor homogeneous . . . nor any of the types I've been able to find for which there are techniques for solving :(

Thank you :P
 
Physics news on Phys.org
You can find an integrating factor for your ODE to make it exact. The general solution in implicit form to your ODE is

x^2+z^2+A^2/(xz)^2= C,

where C is a constant.
 
Furthermore, you can solve the implicit equation for z given by kosostov by means of a clever trick.
 

Similar threads

Undergrad Converting Second Order ODE to Hypergeometric Function
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Question about solving linear first order non-homogeneous ODEs
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Solving Coupled First Order ODEs: Is There a Closed Form Solution?
  • · Replies 28 ·
Replies
28
Views
3K
Graduate Existence of unique solutions to a first order ODE on this interval
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Identifying and solving a pair of ODE's
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad 2nd order ODE's, variation of parameters and the notorious constraint
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Nonlinear Second Order ODE: Can We Find an Analytical Solution?
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Searching for radial part solution of Klein-Gordon type equation
  • · Replies 1 ·
Replies
1
Views
2K
MHB Peak in single ODE within a system
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad ODE Fail: Numerical Solution Oscillations - Possible Solutions | Hi PF
  • · Replies 24 ·
Replies
24
Views
5K