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Boundedness of the solutions of the equation $y'' + x y = 0 $ as $x rightarrow +infty$

Behaviour of Solutions to $x^2y'' + alpha xy'+ beta y = 0$ as $x to 0$ and $x to infty$Modified Airy differential equationHow to show that all solutions of $ay''+by'+cy=0$ approach $0$ as $t rightarrow infty$Existence and Uniqueness of solutions to a differential equation.How do I solve for simple pendulum?Inhomogeneous Airy EquationUnderstanding solutions to the Mathieu Equation in the context of Classical MechanicsBehavior of the solution of the equation $f'' = -x f $ as $xrightarrow infty$Boundedness of Solutions of Differential EquationNonstandard independent solutions of the Airy equation

2

$begingroup$

This is the Airy equation. We know that the two solutions are both bounded as $xrightarrow infty$.

The picture is that the solution just oscillates with higher and higher frequency ($propto sqrtx$) but its amplitude does not increase.

Can we develop this into a general theorem?

Are the solutions of the equation $y''+ x^2 y = 0 $ bounded in both directions?

ordinary-differential-equations

share|cite|improve this question

asked Mar 17 at 23:28

JohnJohn

1874

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Look up perturbation theory.
  $endgroup$
  – MachineLearner
  Mar 17 at 23:30
  

  
  
  
  

add a comment | 

2

$begingroup$

This is the Airy equation. We know that the two solutions are both bounded as $xrightarrow infty$.

The picture is that the solution just oscillates with higher and higher frequency ($propto sqrtx$) but its amplitude does not increase.

Can we develop this into a general theorem?

Are the solutions of the equation $y''+ x^2 y = 0 $ bounded in both directions?

ordinary-differential-equations

share|cite|improve this question

asked Mar 17 at 23:28

JohnJohn

1874

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Look up perturbation theory.
  $endgroup$
  – MachineLearner
  Mar 17 at 23:30
  

  
  
  
  

add a comment | 

$begingroup$

This is the Airy equation. We know that the two solutions are both bounded as $xrightarrow infty$.

The picture is that the solution just oscillates with higher and higher frequency ($propto sqrtx$) but its amplitude does not increase.

Can we develop this into a general theorem?

Are the solutions of the equation $y''+ x^2 y = 0 $ bounded in both directions?

ordinary-differential-equations

share|cite|improve this question

asked Mar 17 at 23:28

JohnJohn

1874

$endgroup$

share|cite|improve this question

asked Mar 17 at 23:28

JohnJohn

1874

share|cite|improve this question

asked Mar 17 at 23:28

JohnJohn

1874

asked Mar 17 at 23:28

JohnJohn

1874

asked Mar 17 at 23:28

JohnJohn

1874