Before solving the equation we are going to see how the solutions behave at
large | y| (and also large | x|, since these variable are proportional!).
For | y| very large, whatever the value of
,
y2, and thus we have to solve
| (7.11) |
Question: Check that these are the solutions. Why doesn't it matter that they don't exactly solve the equations?
Substitute u(y) = H(y)e-y2/2. We find
| (7.12) |
| H''(y) - 2yH'(y) + (2 |
(7.13) |
This equation will be solved by a substitution and infinite series (Taylor series!), and showing that it will have to terminates somewhere, i.e., H(y) is a polynomial!