From Osher Doctorow
The sixth type of Painleve Equation, P_VI, has the form:
1) Dzz(w) = (1/2)u(dw/dz)^2 - vdw/dz + rs
Here r and s, u and v are functions of w and z. Notice that if we
substitute:
2) y = dw/dz
then (1) becomes:
3) dy/dz = (1/2)u y^2 - vy + rs
where rs is a certain function f(w, z). This is somewhat similar to
the Riccati Differential Equation:
4) dy/dt = A(t) + B(t)y + C(t)y^2
except that in (3), B(t) and C(t) are more generally functions of y
and t or of y and z with z replacing t.
PV_IV is somewhat similar to (3) but much simpler, with (1/2)u
replaced by 1/y and -v replaced by 1/z.
Wolfram and Wikipedia have some topics that will help in understanding
122.8, including their versions of the topic "Fundamental Group".
Osher Doctorow
Osher Doctorow
.
|
