| I have just posted in another thread about ordinary differential equation courses being harder than the usual stuff you do in PDE courses. Now I remembered the ultimate example, a problem I never managed to solve. So I'll let you guys have a shot at it.
R'' + (c - r^2)*R = 0. (If you wish, y'' + (c - x^2)*y = 0). The boundary conditions are R'(0) = R(1) = 0. c are the constants for which a solution exists, otherwise known as the Eigenvalues. I know that the lowest Eigenvalue is higher than pi^2 * 0.25 (~2.467) and lower than 2.597. An analytic solution is what I have, however, not been able to find thus far.
I'll tell you what doesn't work : the following changes of function and variables have failed V = R*r, V = R*r^1/2, V = R/r, V = R*sin(r), V = R*cos(r), V = R*exp(r), V = R * exp(r^2), and V = r^n*Bessel(m,r) for which I've tried quite a few combinations of n and m. I'm stuck! Even when I thought I had a solution (not necessarily the lowest Eigenfunction, though) in the form of a series, it didn't seem to fulfill (R'' - r^2*R)/R = constant.
Thanks a bunch to anyone who takes a stab at this! |
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