Download Famous differential equations, and references

FAMOUS SECOND ORDER LINEAR EQUATIONS
In Math 285 we concentrate on constant coefficient equations. But when using separation
of variables to solve heat and wave and Schrödinger equations in disks, balls or in cylinders,
one often ends up at equations with variable coefficients. Here are some of the most famous:
(1 − x2 )y 00 − 2xy 0 + p(p + 1)y = 0 Legendre eq.
(hydrogen atom)
xy 00 + (1 − x)y 0 + py = 0 Laguerre eq.
(hydrogen atom)
2 00
0
2
2
(vibrating membranes)
2 00
0
2
2
(vibrating plates)
x y + xy + (x − p )y = 0 Bessel eq.
x y + xy − (x − p )y = 0 Modified Bessel eq.
y 00 − 2xy 0 + 2py = 0 Hermite eq.
(quantum mechanics)
00
y + xy = 0 Airy eq.
00
(rainbows)
0
x(1 − x)y + [c − (a + b + 1)x]y − aby = 0 Hypergeometric eq.
References for future study
Good books on ordinary differential equations (including the above famous equations)
are
George F. Simmons Differential Equations, with Applications and Historical Notes,
Morris W. Hirsch, Stephen Smale and Robert Devaney Differential Equations, Dynamical
Systems, and an Introduction to Chaos.
A good book on partial differential equations is
Walter A. Strauss Partial Differential Equations: An Introduction.
This is the text for Math 442, and it covers how most of the above famous equations arise
from physically relevant situations. But for a gentle introduction you would be better off
seeing
Stanley J. Farlow Partial Differential Equations for Scientists and Engineers,
Richard Haberman Elementary Applied Partial Differential Equations.
And finally, if you run into a differential equation problem in future, then you are most
welcome to contact me and ask for help!
Richard Laugesen, [email protected]
1