Poincare's theorem and 2nd order ode

In summary, the conversation discusses the second order differential equation y'' + (k^2 + f(r))y(r) = 0, where k is a constant and f(r) is a real valued function with constraints on integratability. According to Poincare's theorem, if a coefficient in the equation depends on a parameter and satisfies certain boundary conditions, the solution is an entire function of the parameter. A reference for this statement can be found in "The Theory of Functions of a Real Variable" by E. T. Copson.
  • #1
jimmy neutron
26
1
This is a request about the second order differential equation
y'' + (k^2 + f(r))y(r) = 0 (1)
where k is a (real) constant and f(r) is a real valued function of r that has some constraints regarding integratability.

According to the following reference:
"Lectures in Scattering Theory", A. G. Sitenko, Pergamon (1971) Chapter 7 pp 86-87:

"According to Poincare's theorem if a coefficient occurring in a second-order differential equation not only depends on the co-ordinate but is also an entire function of some parameter, the solution of the equation, satisfying boundary conditions which are independent of that parameter, is an entire function of the parameter, is an entire function of the parameter for any value of the co-ordinate. Therefore, the function phi(k,r), which is a solution of (1) satisfying the boundary conditions phi(k,0) = 0 and phi'(k,0) = 1 is an entire function of the parameter."

The request is: Can anyone give a reference for this?
 
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  • #2
A reference for the above statement can be found in the following book: "The Theory of Functions of a Real Variable" by E. T. Copson, Dover Publications (1986), Chapter 8, pp. 325-326.
 

FAQ: Poincare's theorem and 2nd order ode

What is Poincare's theorem?

Poincare's theorem is a mathematical theorem that states that if a function has a second order ordinary differential equation (ODE) and satisfies certain conditions, then it has a periodic solution. In other words, the function will repeat itself after a certain interval of time.

What is a second order ODE?

A second order ordinary differential equation (ODE) is a mathematical equation that relates a function to its derivatives. It is a type of differential equation that involves the second derivative of the function. It is commonly used to model physical systems and phenomena in science and engineering.

What are the applications of Poincare's theorem?

Poincare's theorem has applications in various fields such as physics, astronomy, and engineering. It is used to study and analyze periodic systems, predict and understand the behavior of physical systems, and solve problems involving oscillations and vibrations.

What are the conditions for Poincare's theorem to hold true?

There are several conditions that must be satisfied for Poincare's theorem to hold true. These include the function being continuous and bounded, the solution being unique, and the function having a period that is a rational multiple of the period of the driving force.

How is Poincare's theorem related to the concept of chaos?

Poincare's theorem is closely related to the concept of chaos. In chaotic systems, small changes in initial conditions can lead to drastically different outcomes. Poincare's theorem states that in certain conditions, a function can have a periodic solution. However, in chaotic systems, the function does not repeat itself and therefore Poincare's theorem does not apply.

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