The dAlembertian_formal_sol command returns formal solutions with d'Alembertian series coefficients to the given homogeneous linear ordinary differential equation with polynomial coefficients.

If ode is an expression, then it is equated to zero.

The command returns an error message if the differential equation ode does not satisfy the following conditions.

ode must be homogeneous and linear in var

The coefficients of ode must be polynomial in the independent variable of var, for example, $x$, over the rational number field which can be extended by one or more parameters.

A homogeneous linear ordinary differential equation with coefficients that are polynomials in $x$ has a basis of formal solutions (see DEtools[formal_sol]). A formal solution contains a finite number of power series $\sum _{n=0}^{\mathrm{\infty }}v\left(n\right)T^n$ where $T$ is a parameter and the sequence $v\left(n\right)$ satisfies a linear recurrence (homogeneous or inhomogeneous).

This command selects such formal solutions that contain only series with d'Alembertian coefficients. A sequence is called d'Alembertian if it is annihilated by a linear recurrence operator that can be written as a composition of first-order operators (see LinearOperators).

The command determines an integer $N\ge 0$ such that $v\left(n\right)$ can be represented in the form of a d'Alembertian term:

for all $n\ge N$, where $h_i\left(n\right)$, $1\le i\le s\+1$, is a hypergeometric term (see SumTools[Hypergeometric]):

such that $R\left(k\right)=\frac{h_i\left(k+1\right)}{h_i\left(k\right)}$ is rational in $k$ for all $k\ge N$.

'parameter'=T

Specifies the name T that is used to denote $\mathrm{\lambda }{x}^{\frac{1}{r}}$ where $\mathrm{\lambda }$ is a constant and $r$ is called the ramification index. If this option is given, then the command expresses the formal solutions in terms of T and returns a list of lists each of which is of the form [formal solution, relation between T and x]. Otherwise, it returns the formal solutions in terms of $x^{\frac{1}{r}}$.

x=a or 'point'=a

Specifies the expansion point a. It can be an algebraic number, depending rationally on some parameters, or $\mathrm{\infty }$.

The default is $a=0$.

'free'=C

Specifies a base name C to use for free variables C[0], C[1], etc. The default is the global name  _C. Note that the number of free variables may be less than the order of the given equation.

'indices'=[n,k]

Specifies base names for dummy variables. The default values are the global names _n and _k, respectively. The name n is used as the summation index in the power series. the names n1, n2, etc., are used as summation indices in ( + ). The name k is used as the product index in ( ++ ).

'outputHGT'=name

Specifies the form of representation of hypergeometric terms.  The default value is 'inert'.

'inert' - the hypergeometric term ( ++ ) is represented by an inert product, except for $\prod _{k=N}^{n-1}1$, which is simplified to $1$.

'rcf1' or 'rcf2' - the hypergeometric term is represented in the first or second minimal representation, respectively (see ConjugateRTerm).

'active' - the hypergeometric term is represented by non-inert products which, if possible, are computed (see product).

'outputDAT'=name

Specifies the form of representation of the sums in ( + ). The default is 'inert'.

'inert' - the sums are in the inert form, except for trivial sums of the form $\sum _{k=u}^{v-1}1$, which are simplified to $v-u$.

'gosper' - Gosper's algorithm (see Gosper) is used to find a closed form for the sums in ( + ), if possible, starting with the innermost one.

$\mathrm{with}\left(\mathrm{Slode}\right)\:$

$\mathrm{ode}≔\left(-4-x^2+2x\right)y\left(x\right)+\left(2x-3x^3-x^2\right)\mathrm{diff}\left(y\left(x\right)\,x\right)+\left(x^3-x^4\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right)$

$\mathrm{ode}≔\left(-x^2+2x-4\right)y\left(x\right)+\left(-3x^3-x^2+2x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(-x^4+x^3\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)$

$\mathrm{dAlembertian\_formal\_sol}\left(\mathrm{ode}\,y\left(x\right)\,'\mathrm{outputHGT}'='\mathrm{active}'\,'\mathrm{indices}'=\left[n\,k\right]\right)$

$x^2\left(-\frac{\left(\sum _{n=0}^{\mathrm{\infty }}{x}^n\right)}{2}+\frac{\left(\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{\mathrm{n1}=0}^{n-1}{\left(-\frac{1}{2}\right)}^{\mathrm{n1}}\mathrm{\Gamma }\left(\mathrm{n1}+3\right)\left(\mathrm{n1}+2\right)\right)x^n\right)}{4}\right){\mathrm{\_C}}_0+\frac{{\ⅇ}^{\frac{2}{x}}\left(\left(\sum _{n=0}^{\mathrm{\infty }}{x}^n\right)-\frac{1}{3}\right){\mathrm{\_C}}_1}{x}$

$\mathrm{ode}≔{\left(x-1\right)}^2\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)-\left(x-1\right)\left(x-7\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-2\left(2x-5\right)\mathrm{diff}\left(y\left(x\right)\,x\right)-2y\left(x\right)$

$\mathrm{ode}≔{\left(x-1\right)}^2\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)-\left(x-1\right)\left(x-7\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-2\left(2x-5\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-2y\left(x\right)$

$\mathrm{dAlembertian\_formal\_sol}\left(\mathrm{ode}\,y\left(x\right)\,x=0\,'\mathrm{outputHGT}'='\mathrm{inert}'\,'\mathrm{indices}'=\left[n\,k\right]\right)$

${\mathrm{\_C}}_0\left(\sum _{n=0}^{\mathrm{\infty }}{x}^n\right)+{\mathrm{\_C}}_1\left(\sum _{n=0}^{\mathrm{\infty }}n{x}^n\right)+{\mathrm{\_C}}_2\left(\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{\mathrm{n1}=0}^{n-1}\sum _{\mathrm{n2}=0}^{\mathrm{n1}-1}\prod _{k=0}^{\mathrm{n2}-1}\frac{1}{k+3}\right)x^n\right)$