Bessel and Modified Bessel ODEs - Maple Help

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Bessel and Modified Bessel ODEs

Description

• 

The general form of the Bessel ODE is given by the following:

Bessel_ode := x^2*diff(y(x),x,x)+x*diff(y(x),x)+(x^2-n^2)*y(x);

Bessel_ode:=x2ⅆ2ⅆx2yx+xⅆⅆxyx+n2+x2yx

(1)
• 

The general form of the modified Bessel ODE is given by the following:

modified_Bessel_ode := x^2*diff(y(x),x,x)+x*diff(y(x),x)-(x^2+n^2)*y(x);

modified_Bessel_ode:=x2ⅆ2ⅆx2yx+xⅆⅆxyxn2+x2yx

(2)
  

where n is an integer. See Abramowitz and Stegun - `Handbook of Mathematical Functions`, section 9.6.1. The solutions for these ODEs are expressed using the Bessel functions in the following examples.

Examples

withDEtools,odeadvisor

odeadvisor

(3)

odeadvisorBessel_ode

_Bessel

(4)

odeadvisormodified_Bessel_ode

_Bessel,_modified

(5)

The Bessel ODEs can be solved for in terms of Bessel functions:

dsolveBessel_ode

yx=_C1BesselJn,x+_C2BesselYn,x

(6)

dsolvemodified_Bessel_ode

yx=_C1BesselIn,x+_C2BesselKn,x

(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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