The differential equation

Question

𝑦′′
− 𝑥𝑦 = 0
which is also known as Airy’s equation and often encountered in the study of aerodynamics.
Find the series solution of Airy’s equation about ordinary point x = 0. What will be the radius of
Convergence of the series solution.

Answer 1

To find the series solution of Airy's equation about the ordinary point x = 0, we first need to rewrite the equation in terms of x = 0 as the center of our series. Let's define y = Σ(an*x^n), where Σ denotes the sum from n=0 to infinity. Then we have:

y' = Σ(n*an*x^(n-1))
y'' = Σ(n(n-1)*an*x^(n-2))

Plugging these into the Airy's equation, we get:

Σ(n(n-1)*an*x^(n-2)) - x*Σ(an*x^n) = 0
Σ(n(n-1)*an*x^(n-2)) - x*Σ(an*x^n) = Σ(n(n-1)*an*x^(n-2)) - Σ(an*x^(n+1))

We can simplify this expression by matching the exponents of each term, which gives us:

(n*(n-1) - an-1)an - an+2 = 0

This gives us the recurrence relation an+2 = (n*(n-1) - an-1)an. We also have a0 = y(0) and a1 = y'(0) as initial conditions.

By solving this recurrence relation, we can find the coefficients an of the series solution. The series solution will be in terms of the coefficients an and the powers of x.

The radius of convergence of the series solution will be the distance to the nearest singular point of the differential equation, which in this case is x = 0. Therefore, the radius of convergence will be equal to the distance from x = 0 to the nearest singular point, which is 0. Thus, the radius of convergence is 0.