Non-linear second order equation

In summary, the equation y^2 y'' + a y^3 - b = 0, with the given conditions, can be solved using Lie point transformations, resulting in a complicated elliptic integral. The integration process requires finding the roots and evaluating the integral in terms of the elliptic integrals F and \Pi. Despite the efforts of "coomast" and "g_edgar", the integral remains too complex to be solved.
  • #1
Leo Klem
13
0
I am not able to find the general integral of the following non-linear 2nd order equation:

y^2 y'' + a y^3 - b = 0

in which:

y = f(x)

0 < a <= 1, is a constant

b > 0 , is a constant.
 
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  • #2
Hello Leo Klem,

Not an easy one. The only method I know that can be used to solve this equation is Lie point transformations. By setting:

[tex]u=y[/tex]
[tex]v=y'[/tex]
and
[tex]w=\frac{dv}{du}=\frac{y''}{y'}[/tex]

You can transform the equation into:

[tex]v\frac{dv}{du}=\frac{b}{u^2}-au[/tex]

Which has as solution:

[tex]v^2=-\frac{2b}{u}-au^2+K_1[/tex]

Taking the root and inverting the substitution you end up with:

[tex]\pm x+K_2=\int \frac{\sqrt{y}dy}{\sqrt{K_1y-2b-ay^3}}[/tex]

Which is not an easy integral. Maybe by trying to find the roots you can solve it, I did not try this. (check the calculations for errors to be sure)

Hope this helps so far,

coomast
 
  • #3
It's an elliptic integral, not elementary.
Maple evaluates it in terms of the elliptic integrals [itex]F[/itex] and [itex]\Pi[/itex], but too complicated to copy here.
 
  • #4
16 August 2009

Many thanks for the attention paid both by "coomast" and by "g_edgar".
The solution for x given by "ccomast" is just the point where I too had to stop. Unfortunately, I am not in condition to calculate the integral indicated.
Leo Klem
 

1. What is a non-linear second order equation?

A non-linear second order equation is a mathematical expression that involves a second degree polynomial with one or more terms that are not proportional to the variable being squared. In other words, the equation cannot be written in the form of y = ax^2 + bx + c, where a, b, and c are constants.

2. How is a non-linear second order equation different from a linear second order equation?

A linear second order equation can be written in the form of y = ax^2 + bx + c, where a, b, and c are constants. This means that the equation has a constant rate of change and forms a straight line when graphed. A non-linear second order equation, on the other hand, does not have a constant rate of change and forms a curved line when graphed.

3. What are some common examples of non-linear second order equations?

Some common examples of non-linear second order equations include parabolas, circles, ellipses, and hyperbolas. These equations can be used to model a variety of real-world phenomena, such as the trajectory of a projectile, the shape of a satellite's orbit, or the growth of a population.

4. How do you solve a non-linear second order equation?

Solving a non-linear second order equation can be more complex than solving a linear second order equation. Depending on the specific equation, different methods may be used, such as substitution, elimination, or graphing. In some cases, it may not be possible to find an exact solution and approximations or numerical methods may be used.

5. What are some applications of non-linear second order equations in science?

Non-linear second order equations are widely used in various fields of science, including physics, engineering, and economics. They can be used to model and predict the behavior of complex systems, such as fluid dynamics, electrical circuits, and chemical reactions. Non-linear second order equations also play a crucial role in the study of chaos theory and fractals.

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