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exponentialSolutions


Usage

exponentialSolutions L


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


Returns

Returns $[p_1(x,u),\dots,p_n(x,u)]$ such that the exponential solutions of L are described by

\begin{displaymath}
\bigcup_{i=1}^n
\left\{ {\left({\sum_{j=1}^{m_i} c_j f_{ij}(x)}\right)} e^{\int f_{i0}(x) dx} \right\}
\end{displaymath}

where $p_i(x,u) = \sum_{j=0}^{m_i} f_{ij}(x) u^j$ and the $c_j$ are arbitrary constants.


Remarks

exponentialSolutions returns only the solutions that are exponential over $\mathbbm{Q}(x)$, i.e. the solutions $y$ such that $y'/y \in \mathbbm{Q}(x)$. It does not return any eventual solution that is exponential over $\overline{\mathbbm{Q}}(x)$ but not over $\mathbbm{Q}(x)$.


Example

We compute the exponential solutions of

\begin{displaymath}
\frac{d^3 y}{dx^3} - \frac{x^3+1}x \frac{d^2 y}{dx^2}
+ (2x^2-3x) \frac{dy}{dx} + 2 x^4 y = 0
\end{displaymath} (4)

as follows:
1 --> L := D^3-(1+x^3)/x*D^2+(-3*x-2*x^2)*D+2*x^4;
2 --> e := exponentialSolutions(L);
3 --> tex(e);

\begin{displaymath}[ u+x^{2} ]
\end{displaymath}

This means that the exponential solutions of (4) form a one-dimensional vector space generated by $e^{\int x^2 dx}$.


Usage within MAPLE


next up previous contents index
Next: exteriorPower Up: Supported functions Previous: element   Contents   Index
Manuel Bronstein 2002-09-04