The LogarithmicSolution function returns the logarithmic solutions (that is, the solutions in Q(x)[log(x)]) of the specified linear differential system of equations with polynomial coefficients). If such a solution does not exist, then NULL is returned. If the denominator of the solutions cannot be bound then FAIL is returned.

The system parameter is entered either in list form (a list of equations sys and a list of function variables vars to solve for), or in matrix form (matrix A and the independent variable x).

The matrix form specifies the system $\mathrm{Ly}\left(x\right)=\mathrm{Ay}\left(x\right)$, where L is the differential operator, $y\left(x\right)$ is the vector of the functions to solve for, and A is a rational matrix.

For the matrix form of the calling sequence, the case of the system must be specified as 'differential'.

Logarithmic solution is a global problem which is an analog of the local one of finding regular solutions (see LinearFunctionalSystems[RegularSolution]). The logarithmic solutions are of the form $\sum _{m=0}^k{g}_m\left(x\right){\log \left(x\right)}^m$ where the $g_m\left(x\right)$ are rational functions.

The solution involves arbitrary constants of the form _c1, _c2, etc.

The function first computes a universal denominator of the solution using LinearFunctionalSystems[UniversalDenominator]. Then the found denominator is substituted into the system and the resulted system is solved for the solution in Q[x][log(x)], which is performed by an algorithm analogous to one used in LinearFunctionalSystems[RegularSolution].

The error conditions associated with LogarithmicSolution include all those generated by LinearFunctionalSystems[Properties] as well as the following:

The system must be homogeneous.

The case must be differential.

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

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

$\mathrm{sys}≔\left[\mathrm{diff}\left(\mathrm{y1}\left(x\right)\,x\right)x=\mathrm{y2}\left(x\right)\,\mathrm{diff}\left(\mathrm{y2}\left(x\right)\,x\right)=0\right]\:$$\mathrm{vars}≔\left[\mathrm{y1}\left(x\right)\,\mathrm{y2}\left(x\right)\right]$

$\mathrm{vars}≔\left[\mathrm{y1}\left(x\right)\,\mathrm{y2}\left(x\right)\right]$

$\mathrm{vars}≔\left[\mathrm{y1}\left(x\right)\,\mathrm{y2}\left(x\right)\right]\:$

$\mathrm{LogarithmicSolution}\left(\mathrm{sys}\,\mathrm{vars}\right)$

$\left[\ln \left(x\right){\mathrm{\_c}}_1+{\mathrm{\_c}}_2\,{\mathrm{\_c}}_1\right]$

$A≔\mathrm{Matrix}\left(\left[\left[\frac{\frac{1}{x}-x^2}{x^2}\,\frac{x^{-1}}{x^2-1}\right]\,\left[\frac{1}{2x-\frac{1}{x^2}}\,x^4-\frac{2}{x}\right]\right]\right)$

$A≔\left[\begin{array}{cc}\frac{\frac{1}{x}-x^2}{x^2}& \frac{1}{x\left(x^2-1\right)}\\ \frac{1}{2x-\frac{1}{x^2}}& x^4-\frac{2}{x}\end{array}\right]$

$\mathrm{LogarithmicSolution}\left(A\,x\,'\mathrm{differential}'\,\mathrm{hybrid}='\mathrm{false}'\right)$

$\left[0\,0\right]$