Solving the Differential Equation (frac{d^2y}{dx^2} - xy 0) Using Power Series and Airy Functions

In this article, we will explore the solution to the differential equation (frac{d^2y}{dx^2} - xy 0). This is a second-order linear ordinary differential equation that can be approached using both power series methods and special functions, particularly Airy functions.

Steps to Solve the Equation

The given differential equation is:

``` frac{d^2y}{dx^2} - xy 0 ```

We can rearrange it as:

``` frac{d^2y}{dx^2} xy. ```

1. Rearranging the Equation

First, we rearrange the given equation:

``` frac{d^2y}{dx^2} - xy 0 ```

2. Power Series Solution

Assume a solution of the form:

``` y sum_{n0}^{infty} a_n x^n ```

Then, the first and second derivatives are:

``` dy/dx sum_{n1}^{infty} n a_n x^{n-1}, frac{d^2y}{dx^2} sum_{n2}^{infty} n(n-1) a_n x^{n-2} ```

3. Substituting into the Equation

Substitute (y) and (frac{d^2y}{dx^2}) into the original equation:

``` sum_{n2}^{infty} n(n-1) a_n x^{n-2} - x sum_{n0}^{infty} a_n x^n 0 ```

The second term can be rewritten as:

``` x sum_{n0}^{infty} a_n x^n sum_{n1}^{infty} a_{n-1} x^n ```

where (m n - 1).

4. Combining the Series

We can write:

``` sum_{n2}^{infty} n(n-1) a_n x^{n-2} - sum_{m1}^{infty} a_{m-1} x^m 0 ```

Now we can adjust indices to combine the series effectively, aligning the powers of (x).

5. Equating Coefficients

After combining the series, we equate the coefficients of like powers of (x) to create a recurrence relation for (a_n).

6. Special Functions

This equation can also be recognized as related to the Airy differential equation, which has solutions in terms of Airy functions (Ai(x)) and (Bi(x)). The general solution can be expressed as:

``` y(x) C_1 Aileft(frac{x}{sqrt{3}}right) C_2 Bileft(frac{x}{sqrt{3}}right) ```

where (C_1) and (C_2) are constants determined by boundary conditions.

Explanation of Airy Functions

Airy functions are special functions that satisfy the Airy differential equation:

``` frac{d^2 y}{dt^2} - ty 0 ```

The Airy functions are given by:

``` Ai(t) frac{1}{pi} int_{0}^{infty} cosleft(frac{t^{3}}{3} z tright) dz Bi(t) frac{1}{pi} int_{0}^{infty} left[expleft(-frac{t^{3}}{3} z tright) sinleft(frac{t^{3}}{3} z tright) expleft(-frac{t^{3}}{3} - z tright) sinleft(frac{t^{3}}{3} - z tright)right] dz ```

Using the principle of superposition, the general solution to the equation (frac{d^2y}{dx^2} - xy 0) is: