Solving a second-order linear ODE in an infinite limit

In summary, the conversation is about a problem involving the quantum harmonic oscillator and a specific difficulty with a mathematical concept. The given differential equation is y''(x) + (ε-x2) y = 0, and the task is to analyze the behavior of y as x approaches infinity. The solution for y in this limit is expected to be y = A xk e-x2/2. However, when the given solution is substituted into the DE, the "xk" term does not cancel as expected. The conversation suggests taking the limit in the solution and plugging it into the DE to determine the value of k.
  • #1
raving_lunatic
21
0

Homework Statement



So this is part of a broader problem about the quantum harmonic oscillator, but there's one particular bit of mathematics I'm stuck on.

We have the differential equation:

y''(x) +(ε-x2) y = 0

And I'm told that we're to examine how y behaves as x tends towards infinity. I took this to mean that we can ignore the term in epsilon entirely.

We're also told that in this limit, we should obtain y = A xk e-x2/2 as the solution to the differential equation.




Homework Equations





The Attempt at a Solution



I'm not entirely sure how to go about solving the differential equation (bit rusty) but when I substitute in the given solution, the "xk" term doesn't cancel as I suspect that it should - unless of course you just set k = 0, but the next parts of the question require we prefix it with A x^k . I'm confused.
 
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  • #2
raving_lunatic said:

Homework Statement



So this is part of a broader problem about the quantum harmonic oscillator, but there's one particular bit of mathematics I'm stuck on.

We have the differential equation:

y''(x) +(ε-x2) y = 0

And I'm told that we're to examine how y behaves as x tends towards infinity. I took this to mean that we can ignore the term in epsilon entirely.

We're also told that in this limit, we should obtain y = A xk e-x2/2 as the solution to the differential equation.




Homework Equations





The Attempt at a Solution



I'm not entirely sure how to go about solving the differential equation (bit rusty) but when I substitute in the given solution, the "xk" term doesn't cancel as I suspect that it should - unless of course you just set k = 0, but the next parts of the question require we prefix it with A x^k . I'm confused.


You should also take the limit in the solution. What is it approximately equal to? Then plugging in the DE will yield a value of k.
 

FAQ: Solving a second-order linear ODE in an infinite limit

What is a second-order linear ODE?

A second-order linear ODE (ordinary differential equation) is a mathematical equation that relates a function to its first and second derivatives. It can be written in the form of y'' + p(x)y' + q(x)y = g(x), where p(x) and q(x) are functions of x and g(x) is a function of x. The goal in solving a second-order linear ODE is to find the function y(x) that satisfies the equation.

What does it mean to solve a second-order linear ODE in an infinite limit?

Solving a second-order linear ODE in an infinite limit means finding the solution to the equation as x approaches infinity. This can be done by taking the limit of the solution as x goes to infinity, or by using other techniques such as power series expansion or Laplace transforms.

What are the steps to solve a second-order linear ODE in an infinite limit?

The steps to solve a second-order linear ODE in an infinite limit may vary depending on the specific equation, but generally involve finding the general solution to the equation, applying any initial or boundary conditions, and then taking the limit as x goes to infinity. This may also involve using special techniques such as Laplace transforms or power series expansion.

What are some common techniques used to solve a second-order linear ODE in an infinite limit?

Some common techniques used to solve a second-order linear ODE in an infinite limit include taking the limit of the solution as x goes to infinity, using power series expansion, and using Laplace transforms. Other techniques may also be used depending on the specific equation, such as the method of undetermined coefficients or variation of parameters.

Why is solving a second-order linear ODE in an infinite limit important?

Solving a second-order linear ODE in an infinite limit is important because it allows us to understand the behavior of a function as x approaches infinity. This can have practical applications in fields such as physics, engineering, and economics, where understanding the long-term behavior of a system is crucial. Additionally, it is a fundamental concept in mathematics and is used in many areas of study, including differential equations, calculus, and analysis.

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