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hypergeomsols

  

solutions of a hypergeometric type second order linear ODE

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

hypergeomsols(LODE)

hypergeomsols(LODE,y(x))

hypergeomsols(coeff_list,x)

Parameters

LODE

-

homogeneous linear differential equation of second order

y(x)

-

any indeterminate function of one variable; required only when the ODE involves more than one function being differentiated

coeff_list

-

list of coefficients of the linear ODE

x

-

independent variable of the linear ODE

Description

• 

The hypergeomsols routine returns a basis of the space of solutions of a second order linear ODE of hypergeometric type--that is, an equation of the form (see hyperode)

DEtools[hyperode](hypergeom([a,b], [c], x), y(x)) = 0;

yxab+a+b+1xcⅆⅆxyx+x2xⅆ2ⅆx2yx=0

(1)
  

For making symbolic experiments with the hypergeom function and the differential equation it satisfies see also dpolyform.

• 

There are two general forms of calling sequences available for hypergeomsols.

  

The first argument LODE of the first calling sequence is a linear differential equation in diff or D form. The second argument y(x) is the function in the differential equation, and it is required only when the ODE involves more than one function being differentiated.

  

The last calling sequence has as its first argument the list of coefficients of a linear ODE, and the second argument is the independent variable. This input sequence may be convenient for programming with the hypergeomsols routine.

• 

This routine is part of the DEtools package, and so it can be used in the form hypergeomsols(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[hypergeomsols](..).

Examples

withDEtools:

PDEtools[declare]yx,prime=x

yxwill now be displayed asy

derivatives with respect toxof functions of one variable will now be displayed with '

(2)

Kamke's example 2.108:

ode108xⅆ2ⅆx2yx+x+a+bⅆⅆxyx+ayx

ode108xy''+x+a+by'+ay

(3)

B108hypergeomsolsode108,yx

B108KummerMa,a+b,x,KummerUa,a+b,x

(4)

A solution to this ODE is built by using this solution basis and tested as follows.

sol108yx=add_C||iB108i,i=1..2

sol108y=_C1KummerMa,a+b,x+_C2KummerUa,a+b,x

(5)

odetestsol108,ode108

0

(6)

Kamke's example 2.114:

ode114yx+ⅆⅆxyx2x+2+xⅆ2ⅆx2yx

ode114y+y'2x+2+xy''

(7)

B114hypergeomsolsode114,yx

B114ⅇ2xKummerM32,2,2x,ⅇ2xKummerU32,2,2x

(8)

Kamke's example 2.116:

ode116xⅆ2ⅆx2yx+ax+b+nⅆⅆxyx+nayx

ode116xy''+ax+b+ny'+nay

(9)

B116hypergeomsolsode116,yx

B116KummerMn,b+n,ax,KummerUn,b+n,ax

(10)

See Also

DEtools

dpolyform

dsolve

hyperode