Valid Series Solution for Airy's Equation

What is the valid series solution for Airy's equation?

Valid Series Solution for Airy's Equation

When solving Airy's equation y" - xy = 0, we can obtain a valid series solution using the power series method. By assuming a power series solution of the form y(x) = Σaₙxⁿ, where aₙ are the coefficients to be determined, we can proceed with the following steps:

1. Differentiate y(x) twice to find y''(x) = Σaₙn(n-1)xⁿ⁻².

2. Substitute the expressions into the Airy's equation to get Σaₙn(n-1)xⁿ⁻² - xΣaₙxⁿ = 0.

3. Simplify the equation and collect terms with the same power of x to obtain Σaₙ(n(n-1) - x)aₙxⁿ⁻² = 0.

4. Since this equation holds for all x, each term in the summation must be equal to zero, leading to aₙ(n(n-1) - x) = 0.

5. Solve the characteristic equation n(n-1) - x = 0 to find roots n₁ = 0 and n₂ = 1.

6. Write the general solution as y(x) = c₁Ai(x) + c₂Bi(x), where Ai(x) and Bi(x) are the Airy functions, and c₁ and c₂ are arbitrary constants determined by initial conditions.

By following these steps, we can derive a valid series solution for Airy's equation, ensuring accurate and reliable results.