Complete Elliptic Integral ODEs - Maple Help

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Complete Elliptic Integral ODEs

Description

• 

The general forms of the elliptic ODEs are given by the following:

elliptic_I_ode := diff(x*(1-x^2)*diff(y(x),x),x)-x*y(x)=0;

elliptic_I_ode:=x2+1ⅆⅆxyx2x2ⅆⅆxyx+xx2+1ⅆ2ⅆx2yxxyx=0

(1)

elliptic_II_ode := (1-x^2)*diff(x*diff(y(x),x),x)+x*y(x)=0;

elliptic_II_ode:=x2+1ⅆⅆxyx+xⅆ2ⅆx2yx+xyx=0

(2)
  

See Gradshteyn and Ryzhik, "Tables of Integrals, Series and Products", p. 907. The solution to this type of ODE can be expressed in terms of the EllipticK and EllipticCK functions.

Examples

withDEtools,odeadvisor

odeadvisor

(3)

dsolveelliptic_I_ode

yx=_C1EllipticKx+_C2EllipticCKx

(4)

dsolveelliptic_II_ode

yx=_C1EllipticEx+_C2EllipticCExEllipticCKx

(5)

odeadvisorelliptic_I_ode

_elliptic,_class_I

(6)

odeadvisorelliptic_II_ode

_elliptic,_class_II

(7)

See Also

DEtools, odeadvisor, dsolve, and ?odeadvisor,<TYPE> where <TYPE> is one of: quadrature, missing, reducible, linear_ODEs, exact_linear, exact_nonlinear, sym_Fx, linear_sym, Bessel, Painleve, Halm, Gegenbauer, Duffing, ellipsoidal, elliptic, erf, Emden, Jacobi, Hermite, Lagerstrom, Laguerre, Liouville, Lienard, Van_der_Pol, Titchmarsh; for other differential orders see odeadvisor,types.


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