As usual there is a snag. Most equations of interest are of a form where p and/or q are singular at the point {t}_{0} (usually {t}_{0} = 0). Any point {t}_{0} where p(t) and q(t) are singular is called (surprise!) a singular point. Of most interest are a special class of singular points called regular singular points, where the differential equation can be given as
|
{(t − {t}_{0})}^{2}y''(t) + (t − {t}_{
0})α(t)y'(t) + β(t)y(t) = 0,
| (9.10) |
with α and β analytic at t = {t}_{0}. Let us assume that this point is {t}_{0} = 0. Frobenius’ method consists of the following technique: In the equation
|
{x}^{2}y''(x) + xα(x)y'(x) + β(x)y(x) = 0,
| (9.11) |
we assume a generalised series solution of the form
|
y(x) = {x}^{γ}{ \mathop{∑
}}_{n=0}^{∞}{c}_{
n}{x}^{k}.
| (9.12) |
Equating powers of x we find
|
γ(γ − 1){c}_{0}{x}^{γ} + {α}_{
0}γ{c}_{0}{x}^{γ} + {β}_{
0}{c}_{0}{x}^{γ} = 0,
| (9.13) |
etc. The equation for the lowest power of x can be rewritten as
|
γ(γ − 1) + {α}_{0}γ + {β}_{0} = 0.
| (9.14) |
This is called the indicial equation. It is a quadratic equation in γ, that usually has two (complex) roots. Let me call these {γ}_{1}, {γ}_{2}. If {γ}_{1} − {γ}_{2} is not integer one can prove that the two series solutions for y with these two values of γ are independent solutions.
Let us look at an example
|
{t}^{2}y''(t) + {3\over
2}ty'(t) + ty = 0.
| (9.15) |
Here α(t) = 3∕2, β(t) = t, so t = 0 is indeed a regular singular point. The indicial equation is
|
γ(γ − 1) + {3\over
2}γ = {γ}^{2} + γ∕2 = 0.
| (9.16) |
which has roots {γ}_{1} = 0, {γ}_{2} = −1∕2, which gives two independent solutions
Independent solutions:
Independent solutions are really very similar to independent vectors: Two or more functions
are independent if none of them can be written as a combination of the others. Thus
x and
1 are independent,
and 1 + x and
2 + x are
dependent.