Solving the Ode Emden-Fowler Equation: A Scientist's Perspective

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In summary, the conversation discusses a particular solution for the differential equation y''=y^2/x, which is a special case of Kamke's differential equation 6.73. The solution is given as y(x)=2/x, and it is also mentioned that Maple gives a complicated implicit solution. Additionally, there is a mention of Erich Kamke, a German mathematician, and his book about the equation.
  • #1
kamke
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Hi,
y'' = y^2/x
y(x) = ?
Trial and error: y(x) = 2/x.
I am glad to get a particular solution too.
Thanks,
kamke
 
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  • #2
Dear Kamke,

You of all people should know that this is a special case of Kamke's differential equation 6.73! But indeed, when you wrote the book, there wasn't much known about this equation.
Here's what is said in the book, translated in English:
..
[itex]u^{''}=y^n x^{1-n}[/itex]
for the special case that n=2, one solution is 2/x^2

Well, I am glad that after 65 years, you've managed to correct your mistake.

By the way, also Maple gives a very complicated implicit solution, so I can't help you further.
 
  • #3
Dear Bigfooted,

Thanks.

Erich Kamke (1890-1961) German mathematician
u'' = x^(1-n)*u^n
n = 2
u'' = x^(-1)*u^2
u(x) = 2/x (+ u ≡ 0)
I take great interest in implicit solution.

kamke
 

What is the Ode Emden Fowler equation?

The Ode Emden Fowler equation is a second-order ordinary differential equation that is used to describe the structure of polytropic gas spheres. It is commonly used in astrophysics and fluid mechanics.

What is the significance of the Ode Emden Fowler equation?

The Ode Emden Fowler equation is significant because it provides a mathematical model for the density distribution of a self-gravitating, spherically symmetric, polytropic gas sphere. This allows scientists to better understand the behavior and structure of celestial bodies, such as stars and planets.

Who developed the Ode Emden Fowler equation?

The Ode Emden Fowler equation was developed by German mathematician Karl Wilhelm Emden and British mathematician Alfred Fowler in the early 20th century.

What is a polytropic gas sphere?

A polytropic gas sphere is a gaseous body in which the pressure and density are related by a power law, where the exponent is known as the polytropic index. This type of gas sphere is often used to model the structure of stars and other celestial bodies.

How is the Ode Emden Fowler equation used in astrophysics?

The Ode Emden Fowler equation is used in astrophysics to model the internal structure and evolution of stars. By solving this equation, scientists can determine the density distribution, temperature, and other important properties of a star at different stages of its life cycle.

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