# Solution of Airy's Equation in Terms of Gauss Hypergeometric Series

• Ted123
In summary: F_1\left(1,\frac{2}{3};9;z^3 \right) + a_1 \;_2F_1\left(1,\frac{4}{3};9;z^3 \right)In summary, the general solution of Airy's equation satisfying the initial conditions f(0)=1, f'(0)=0 can be expressed in terms of the Gauss hypergeometric series as f(z) = a_0 \;_2F_1\left(1,\frac{2
Ted123

## Homework Statement

Find the general solution of Airy's equation $$f'' - zf=0$$ satisfying the initial conditions f(0)=1, f'(0)=0 as a power series expansion at z=0. Express the result in terms of the Gauss hypergeometric series.

## The Attempt at a Solution

After subbing $$f(z)=\sum_{n=0}^{\infty} a_n z^n$$ into Airy's equation and manipulating the summations I get the recurrence relation: $$a_2 =0$$ $$a_{n+3} = \frac{a_{n}}{(n+2)(n+3)}\;,\;\;\;\forall \;\;n=0,1,2,3,...$$
Solving this:
$$a_{3k+2}=0\;\;\;\forall \;\;k=0,1,2,3,...$$ and solving separately for n=3k and n=3k+1, $$a_{3(k+1)} = \frac{a_{3k}}{(3k+3)(3k+2)} = \frac{a_{3k}}{9(k+1)(k+\frac{2}{3})}$$ $$a_{3k} = \frac{a_0}{9^k (1)_k (\frac{2}{3})_k}$$ Similarly for n=3k+1, $$a_{3k+1} = \frac{a_1}{9^k (1)_k (\frac{4}{3})_k}$$ so that the solution is $$f(z) = a_0 \left( \sum_{k=0}^{\infty} \frac{1}{9^k (1)_k (\frac{2}{3})_k} z^{3k} \right) + a_1 \left( \sum_{k=0}^{\infty} \frac{1}{9^k (1)_k (\frac{4}{3})_k} z^{3k+1} \right)$$

f(0)=a0, f'(0)=a1

Hence the solution to the initial value problem f(0)=1, f'(0)=0 is:

$$f(z) = \sum_{k=0}^{\infty} \frac{1}{9^k (1)_k (\frac{2}{3})_k} z^{3k} = \sum_{k=0}^{\infty} \frac{1}{9^k (\frac{2}{3})_k} \frac{z^{3k}}{k!}$$

How do I express this in terms of the Gauss hypergeometric series?:
[PLAIN]http://img200.imageshack.us/img200/5992/gauss.png

Last edited by a moderator:

Great job on finding the general solution to Airy's equation and satisfying the initial conditions! To express the solution in terms of the Gauss hypergeometric series, we can use the identity:
_2F_1(a,b;c;z) = \sum_{k=0}^{\infty} \frac{(a)_k (b)_k}{(c)_k} \frac{z^k}{k!}
where (x)_k is the Pochhammer symbol.
In this case, we can rewrite the solution as:
f(z) = a_0 \left( \sum_{k=0}^{\infty} \frac{(1)_k (\frac{2}{3})_k}{(9)_k} z^{3k} \right) + a_1 \left( \sum_{k=0}^{\infty} \frac{(1)_k (\frac{4}{3})_k}{(9)_k} z^{3k+1} \right)
= a_0 \left( \sum_{k=0}^{\infty} \frac{(1)_k (\frac{2}{3})_k}{(9)_k} z^{3k} \right) + a_1 \left( \sum_{k=0}^{\infty} \frac{(1)_k (\frac{4}{3})_k}{(9)_k} z^{3k} \frac{z}{k+1} \right)
= a_0 \left( \sum_{k=0}^{\infty} \frac{(1)_k (\frac{2}{3})_k}{(9)_k} z^{3k} \right) + a_1 \left( \sum_{k=1}^{\infty} \frac{(1)_{k-1} (\frac{4}{3})_{k-1}}{(9)_{k-1}} z^{3k} \frac{z}{k} \right)
= a_0 \left( \sum_{k=0}^{\infty} \frac{(1)_k (\frac{2}{3})_k}{(9)_k} z^{3k} \right) + a_1 \left( \sum_{k=0}^{\infty

## 1. What is the Airy's equation?

The Airy's equation is a second-order linear differential equation that is commonly used in physics and engineering to describe the behavior of physical systems involving oscillations and decay. It is written in the form of y''(x) + xy(x) = 0, where y(x) is the unknown function.

## 2. What is the Gauss Hypergeometric series?

The Gauss Hypergeometric series is a special type of infinite series that can be used to represent solutions to certain types of differential equations, including the Airy's equation. It is written in the form of F(a, b; c; x) = 1 + (a*b)/c * x + (a*(a+1)*b*(b+1))/(c*(c+1)) * x^2 + ..., where a, b, and c are constants and x is the variable.

## 3. How is the Airy's equation solved using the Gauss Hypergeometric series?

The Airy's equation can be transformed into a form that can be solved using the Gauss Hypergeometric series by making a change of variable and using the Frobenius method. This results in a solution in terms of the Gauss Hypergeometric series, which can be evaluated to obtain the desired solution.

## 4. What are the applications of the solution of Airy's equation in terms of Gauss Hypergeometric series?

The solution of Airy's equation in terms of Gauss Hypergeometric series has various applications in physics and engineering, including in the study of quantum mechanics, fluid mechanics, and electromagnetism. It is also used in the analysis of systems with oscillatory and decaying behavior, such as in the study of damped harmonic oscillators and heat conduction.

## 5. Are there any limitations to the solution of Airy's equation in terms of Gauss Hypergeometric series?

While the solution of Airy's equation in terms of Gauss Hypergeometric series is a powerful tool for solving certain types of differential equations, it is not applicable to all types of equations. Additionally, it may be difficult to evaluate the series for certain values of the constants, and numerical methods may be necessary in those cases.

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