Efficient Solutions for Solving First Order ODE with Constant A

  • Thread starter BobbyBear
  • Start date
  • Tags
    Ode
In summary, the conversation is about solving a first order ordinary differential equation with a given form, where A is a constant. The equation is neither exact nor homogeneous, and the person is looking for techniques to solve it. Another person suggests finding an integrating factor to make it exact, and provides the general solution in implicit form. They also mention a clever trick for solving the implicit equation for z.
  • #1
BobbyBear
162
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
  • #2
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.
 
  • #3
Furthermore, you can solve the implicit equation for z given by kosostov by means of a clever trick.
 

FAQ: Efficient Solutions for Solving First Order ODE with Constant A

What is a 1st order ODE?

A 1st order ODE (ordinary differential equation) is a mathematical equation that describes the relationship between an unknown function and its derivatives. It involves only first-order derivatives and can be written in the form: dy/dx = f(x,y).

Why do we need help with 1st order ODEs?

ODEs are used to model many real-world phenomena in fields such as physics, biology, and engineering. Solving them can be challenging and may require specialized techniques, so seeking help can be beneficial.

What are some common methods for solving 1st order ODEs?

Some common methods include separation of variables, variation of parameters, and integrating factors. Other techniques include Laplace transforms, power series solutions, and numerical methods.

Can 1st order ODEs be solved analytically?

Yes, if the equation is simple enough, it can be solved analytically using one of the aforementioned methods. However, for more complex equations, numerical methods may be necessary.

How can I check if my solution to a 1st order ODE is correct?

You can check your solution by plugging it back into the original equation and verifying that it satisfies the equation. You can also use software or online tools to graph and compare your solution to the actual solution.

Similar threads

I Second Order ODE with Exponential Coefficients
Replies
1
Views
2K
I Converting Second Order ODE to Hypergeometric Function
Replies
7
Views
2K
I Question about solving linear first order non-homogeneous ODEs
Replies
3
Views
2K
Solving Coupled First Order ODEs: Is There a Closed Form Solution?
Replies
28
Views
2K
A Existence of unique solutions to a first order ODE on this interval
Replies
2
Views
1K
I How to do Magnus expansion for time-dependent companion matrix?
Replies
7
Views
976
A Identifying and solving a pair of ODE's
Replies
6
Views
1K
I Nonlinear Second Order ODE: Can We Find an Analytical Solution?
Replies
2
Views
3K
MHB Peak in single ODE within a system
Replies
5
Views
2K
I Solving an ODE with Legendre Polynomials
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top