Bessel and Modified Bessel ODEs
Description
Examples
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
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+xⅆⅆxyx−n2+x2yx
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.
withDEtools,odeadvisor
odeadvisor
odeadvisorBessel_ode
_Bessel
odeadvisormodified_Bessel_ode
_Bessel,_modified
The Bessel ODEs can be solved for in terms of Bessel functions:
dsolveBessel_ode
yx=c__1BesselJn,x+c__2BesselYn,x
dsolvemodified_Bessel_ode
yx=c__1BesselIn,x+c__2BesselKn,x
See Also
DEtools
dsolve
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
odeadvisor,types
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