Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

234

5.4.1 Bessel equation

One of the more important equations in mathematical applications is the so called Bessel or cylindrical equation L ( y ) ≡ x 2 y ′′ + xy ′ +( x 2 − ν 2 ) y = 0 , (5.45) where ν is a constant (Re ν > 0). This equation occurs in problems of oscillations, electrostatic fields, heat conduction, etc. We will further assume that ν is a real parameter. Let us demonstrate the procedure described above on this equation. First, we can observe a regular singularity at point x = 0. In this case, the index equation λ ( λ − 1 )+ 1 · λ +( 0 − ν 2 )= 0 is a quadratic equation λ 2 − ν 2 = 0 , (5.46) with the following solutions λ 1 = ν , λ 2 = − ν . Let us assume that ν = 0, from where it follows that λ 1 = λ 2 = 0. According to Theorem 15 b), p.233, in this case we obtain the following solutions y 1 ( x )= | x | 0 r 1 ( x ) y 2 ( x )= | x | 0 + 1 r 2 ( x )+ y 1 ( x ) log | x | , or if we observe the case where x > 0

y 1 ( x )= r 1 ( x ) y 2 ( x )= xr 2 ( x )+ y 1 ( x ) log x .

According to the initial assumptions, r i ( x ) are analytical functions for x = 0, and each of them can thus be represented by a series that converges for all finite values of x . Let us first determine r 1 r 1 ( x )= ∞ ∑ k = 0 a k x k , a 0̸ = 0 (5.47) and calculate L ( r 1 ) . Given that

∞ ∑ k = 1

∞ ∑ k = 2

k − 1 i r ′′

k − 2 ,

r ′ 1 ( x )=

ka k x

1 ( x )=

k ( k − 1 ) a k x

(5.48)

it follows that

∞ ∑ k = 2

∞ ∑ k = 1

∞ ∑ k = 0

L ( r 1 ) ≡ x 2

k − 2 + x

k − 1 + x 2

k = 0 ,

k ( k − 1 ) a k x

ka k x

a k x

(5.49)

being one of the solutions ( y 1 ) of the initial equation.