Jacobi Elliptic Functions - Jacobi Elliptic Functions As Solutions of Nonlinear Ordinary Differential Equations

Jacobi Elliptic Functions As Solutions of Nonlinear Ordinary Differential Equations

The derivatives of the three basic Jacobi elliptic functions are:

\frac{\mathrm{d}}{\mathrm{d}z}\, \mathrm{sn}\,(z) = \mathrm{cn}\,(z)\, \mathrm{dn}\,(z),



\frac{\mathrm{d}}{\mathrm{d}z}\, \mathrm{dn}\,(z) = - k^2 \mathrm{sn}\,(z)\, \mathrm{cn}\,(z).

With the addition theorems above and for a given k with 0 < k < 1 they therefore are solutions to the following nonlinear ordinary differential equations:

  • solves the differential equations
and
  • solves the differential equations
and
  • solves the differential equations
and

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