A second order nonlinear ordinary differential equation

Click For Summary

Discussion Overview

The discussion revolves around the methods for solving a second order nonlinear ordinary differential equation of the form x''x = ((x')^2)/2. Participants explore various approaches, including guessing functions, substitutions, and numerical solutions, while addressing the challenges posed by the equation's nonlinearity.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests guessing a function that satisfies the differential equation, specifically one that behaves like x' = a x, where a is a constant.
  • Another participant proposes finding a pair of real numbers a and b such that the derivative d/dt (x^a x'^b) is proportional to the left-hand side of the equation.
  • A different approach is mentioned, where the equation is classified as an autonomous system, allowing for a substitution of x'' = x' dx'/dx to facilitate integration.
  • One participant inquires whether a numerical solution is sufficient given specific boundary conditions.
  • A rearrangement of the equation is presented, leading to a quadrature form that may assist in solving the equation.

Areas of Agreement / Disagreement

Participants express various methods for approaching the problem, indicating a lack of consensus on a single solution strategy. Multiple competing views on how to tackle the equation remain present.

Contextual Notes

Some methods rely on specific assumptions about the form of the solution or the nature of the equation, which may not be universally applicable. The discussion does not resolve the mathematical steps involved in the proposed solutions.

Who May Find This Useful

Individuals interested in nonlinear ordinary differential equations, mathematical modeling, or those seeking various methods for solving complex differential equations may find this discussion relevant.

Lightfuzz
Messages
15
Reaction score
0
How, if possible, could I solve the equation: x''x=((x')^2)/2? Thanks.
 
Physics news on Phys.org
If you are allowed to guess, then try guess a very often used function that when differentiated gives a scaled version of itself, i.e. a function that satisfy x' = a x, with a being a constant.
 
Try to find such a pair of real numbers a & b that d/dt (x^a x'^b) is proportional to x"x-((x')^2)/2.
 
filiplarsen said:
If you are allowed to guess, then try guess a very often used function that when differentiated gives a scaled version of itself, i.e. a function that satisfy x' = a x, with a being a constant.

I've tried that but is there a way to actually solve it without guessing?
 
Equations like this, which do not contain the independent variable, are called autonomous systems. They are easily solved with the substitution x'' = x' dx'/dx . Then collect the terms in x' and x and integrate them both. This will give you x' in terms of x, which you can then integrate a second time.
 
Do you only require a numerical solution given some particular boundary conditions?
 
rearrange:

\frac{1}{2x}=\frac{\ddot{x}}{x}=-\frac{d}{dx}(\frac{1}{\dot{x}})

Then, reduce to a quadrature:

dt=(C-ln(\sqrt{x}))dx =d[x(C'-ln(\sqrt{x}))]
 

Similar threads

Undergrad Why Is a Differential Equation Called Nonlinear?
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Solving nonlinear singular differential equations
  • · Replies 3 ·
Replies
3
Views
2K
Graduate FEM basis polynomial order and the differential equation order
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Second order non-homogeneous linear ordinary differential equation
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Applying the Laplace transform to solve Differential equations
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Question about the general solution to Hooke's law
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Solving Differential Equation Using Reduction of Order
  • · Replies 2 ·
Replies
2
Views
4K
Undergrad Second order ordinary differential equation to a system of first order
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad How do I solve this first order second degree differential equation?
  • · Replies 7 ·
Replies
7
Views
4K
Undergrad Number of solutions to the order of the equation
  • · Replies 2 ·
Replies
2
Views
3K