radicalSolutions

Usage

radicalSolutions L
radicalSolutions(L,m)

Parameter	Type	Description
L	$\mathbbm{Q}[x,\frac d{dx}]$	A differential operator
m	$\mathbbm{Z}$	A positive integer

Returns

Returns a matrix

$$\pmatrix{ p_1(x,u) & f_{11}(x) & \cdots & f_{1n}(x) \cr p_2(... ...s & \vdots \cr p_s(x,u) & f_{s1}(x) & \cdots & f_{sn}(x) \cr }$$

such that the radical solutions of L are described by

$$\bigcup_{i=1}^s \left\{ {\left({\sum_{j=1}^n c_j f_{ij}(x)}\right)} \prod_{j=0}^{(n_j-1)/2} p_{i,2j+1}(x)^{p_{i,2j}} \right\}$$

where $p_i(x,u) = \sum_{j=0}^{n_j} p_{ij}(x) u^j$ and the $c_j$ are arbitrary constants. If a second argument $m > 0$ is present, then only the solutions $y$ such that $y^m \in \mathbbm{Q}(x)$ are returned.

Remarks

radicalSolutions returns only the solutions that are radical over $\mathbbm{Q}$, i.e. the solutions $y$ such that $y^e \in \mathbbm{Q}(x)$ for some integer $e > 0$. It does not return any eventual solution that is radical over $\overline{\mathbbm{Q}}$ but not over $\mathbbm{Q}$.

Example

We compute the radical solutions of

$$16 \frac {d^2 y}{dx^2} + \frac{3}{x^2} y = 0$$ (7)

as follows:

1 --> L := 16*D^2 + 3/x^2;
2 --> r := radicalSolutions(L);
3 --> tex(r);

$$\pmatrix{ x\,u+{{1} \over {4}} & 1 \cr x\,u+{{3} \over {4}} & 1\cr }$$

This means that the radical solutions of (7) form a two-dimensional vector space over the constants generated by $x^{1/4}$ and $x^{3/4}$.

Usage within MAPLE

• When using radicalSolutions from inside MAPLE, the result returned from BERNINA is transformed into actual radicals, so the above example in MAPLE would be:

  > L := 16*D^2 + 3/x^2;
  > radicalSolutions(L, D, x);
  

  
  
  $$\left[{x}^{3/4},\sqrt [4]{x}\right]$$
  
  

  
  See Also

  exponentialSolutions