Solving a second-order linear ODE in an infinite limit

[raving_lunatic]

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 x^k 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 "x^k" 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.

 

[nrqed]

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 x^k 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 "x^k" 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.