d²y/dx²-xy=0 (Airy differential equation) is solved by writing y as a series in x:
y(x)=∑aᵣxʳ for r ∊ [0,∞), where aᵣ is the general coefficient to be found, that is:
y=a₀+a₁x+a₂x²+a₃x³+a₄x⁴+...+aᵣxʳ.
So dy/dx=a₁+2a₂x+3a₃x²+4a₄x³+...+raᵣxʳ⁻¹, and
d²y/dx²=2a₂+6a₃x+12a₄x²+...+r(r-1)aᵣxʳ⁻².
xy=a₀x+a₁x²+a₂x³+a₃x⁴+a₄x⁵+...+aᵣxʳ⁺¹.
d²y/dx²-xy=2a₂+6a₃x+12a₄x²+...+r(r-1)aᵣxʳ⁻²-(a₀x+a₁x²+a₂x³+a₃x⁴+a₄x⁵+...+aᵣxʳ⁺¹).
If the terms are arranged according to powers of x we can write this in tabular form
| constant |
x |
x² |
x³ |
x⁴ |
xʳ |
| 2a₂ |
6a₃-a₀ |
12a₄-a₁ |
20a₅-a₂ |
30a₆-a₃ |
(r+1)(r+2)aᵣ₊₂-aᵣ₋₁ |
From this it follows that y=2a₂+(6a₃-a₀)x+(12a₄-a₁)x²+...+((r+1)(r+2)aᵣ₊₂-aᵣ₋₁)xʳ, which can be written:
2a₂+∑((r+1)(r+2)aᵣ₊₂-aᵣ₋₁)xʳ for r ∊ [1,∞)=0, to represent d²y/dx²-xy=0.
For this to be true, 2a₂=0, so a₂=0, and (r+1)(r+2)aᵣ₊₂-aᵣ₋₁=0, and aᵣ₊₂=aᵣ₋₁/((r+1)(r+2)).
When r-1=2, r=3, so a₅=a₂/20, but a₂=0, so a₅=0=a₈=a₁₁=..., that is all aᵣ where r=3n+2 for integer n≥0.
We can express all other coefficients in terms of a₀ or a₁ by applying the recursion formula:
a₃=a₀/6, a₆=a₃/30=(a₀/6)/30, a₉=a₆/72=((a₀/6)/30)/72, etc.
a₄=a₁/12, a₇=a₄/42=(a₁/12)/42, a₁₀=a₇/90=((a₁/12)/42)/90, etc.
The solution for y is therefore:
y=a₀(1+x³/6+x⁶/180+x⁹/12960+...)+a₁(x+x⁴/12+x⁷/504+x¹⁰/45360+...).
This can also be written: y=a₀y₁+a₁y₂ where:
y₁=1+x³/6+x⁶/180+x⁹/12960+...
and
y₂=x+x⁴/12+x⁷/504+x¹⁰/45360+...
There are various ways of representing the coefficients for y₁ and y₂, for example, for y₁ we have (2×3)(5×6)(8×9)(...) as the denominator, and for y₂ (3×4)(6×7)(9×10)(...) as the denominator.