polynomialSolution

Usage

polynomialSolution(L, g)

Parameter	Type	Description
L	$\mathbbm{Q}[x,\frac d{dx}]$	A differential operator
g	$\mathbbm{Q}[x]$	A polynomial

Description

polynomialSolution($L,g$) returns either $[p]$ where $p \in \mathbbm{Q}[x]$ satisfies $L p = g$, or $[]$ if $L y = g$ has no solution in $\mathbbm{Q}[x]$.

Remarks

polynomialSolution($L,0$) returns $[0]$ only when $L y = 0$ has no nonzero polynomial solution.

Example

A particular polynomial solution of

$$x^3 \frac{d^3y}{dx^3}+x^2 \frac{d^2y}{dx^2}-2x \frac{dy}{dx} +2 y = 2 x^4$$

can be computed in the following way:

1 --> L := x^3*D^3+x^2*D^2-2*x*D+2;
2 --> p := polynomialSolution(L,2*x^4);
3 --> tex(p);

$$[ {{1} \over {15}}\,x^{4} ]$$

Usage within MAPLE

• When using polynomialSolution from inside MAPLE, the output is modified and either a polynomial solution in $p \in \mathbbm{Q}[x]$ or $[]$ is returned. So the above example in MAPLE would be:

  > L := x^3*D^3+x^2*D^2-2*x*D+2;
  > polynomialSolution(L,2*x^4,D,x);
  

  
  
  $${{1} \over {15}}\,x^{4}$$
  
  

  
  See Also

  kernel, rationalSolution