A Second Order Differential Equation

[cyunus]

Homework Statement

Hi, this problem is from first chapter of Mathematical Methods of Physics by Mathews and Walker. (Problem 1-36, second edition)

Consider the differential equation $y'' - xy + y^3 = 0$ for large positive x.

a-) Find an oscillating solution with two arbitrary constants.

b-) Find a (non trivial) particular nonoscillating solution.

Homework Equations

$x \rightarrow \infty$ $Ai(x) \rightarrow \frac{e^{-\frac{2}{3}x^{\frac{3}{2}}}}{2 \sqrt{\pi}x^{\frac{1}{4}}}$ and $Bi(x) \rightarrow \frac{e^{\frac{2}{3}x^{\frac{3}{2}}}}{\sqrt{\pi}x^{\frac{1}{4}}}$

The Attempt at a Solution

If we assume y is small then we get airy equation and $Ai(x)$ is a solution while $Bi(x)$ is not a solution since for $x \rightarrow \infty { } Bi(x) \rightarrow \infty$. Also it should be noted that $Ai(x)$ does not have a root for large x therefore cannot be considered as an oscillated solution. Also WKB method didn't give me an oscillating solution but I can't say I know it well.

Thanks

 

[mighty2000]

for the post! I agree with your approach so far. For part a), you are correct that Ai(x) does not have a root for large x, so it cannot be considered an oscillating solution. However, there is another solution to the Airy equation that does oscillate: Bi(x). This is the solution that you mentioned in your post, but there is one small mistake. For large x, Bi(x) does not go to infinity, it actually oscillates between positive and negative values. So, for part a), you can use Bi(x) as your oscillating solution and add two arbitrary constants to it to get a general solution.

For part b), you are correct that Ai(x) and Bi(x) are not solutions for large x, but there is another approach you can take. You can use the fact that for large x, the term y'' becomes negligible compared to the other terms in the equation. This means you can approximate the equation as xy - y^3 = 0. Solving this equation for y will give you a non-trivial particular solution. Can you try this approach and see if you can find a solution?

As for the WKB method, it is a powerful tool for finding solutions to differential equations, but it may not be necessary for this problem. However, if you are interested in learning more about it, I recommend looking into it further.