The HomogeneousSystem function returns the homogeneous system that is equivalent to the given linear functional system of equations with polynomial coefficients.

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, vector b, and the independent variable x, where the vector b is optional).

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

For the matrix from of the calling sequence, the case of the system must be specified as one of 'differential', 'difference', or 'qdifference'. If b is not specified, the system is assumed to be homogeneous.

The parameter homo is set to true if the given system is homogeneous and false otherwise.

All of the base functions of the LinearFunctionalSystems package are designed to handle homogeneous systems. In order to solve inhomogeneous systems, the following technique is used.

For a given inhomogeneous system, HomogeneousSystem constructs the equivalent homogeneous system by adding one more equation for the constant function and transforming all right-hand sides of the given system into the coefficients of this new function.

The equivalent homogeneous system that is returned is in the form suitable to be used by other functions of the LinearFunctionalSystems package. Namely, it either returns a sequence of a new list of equations and a list of functions to solve for, or a new matrix, vector, variable, and case.

The error conditions associated with HomogeneousSystem are the same as those which are generated by LinearFunctionalSystems[Properties].

This function is part of the LinearFunctionalSystems package, and so it can be used in the form HomogeneousSystem(..) 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[HomogeneousSystem](..).

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

$\mathrm{sys}≔\left[\mathrm{diff}\left(\mathrm{y1}\left(x\right)\,x\right)-\mathrm{y2}\left(x\right)\,\mathrm{diff}\left(\mathrm{y2}\left(x\right)\,x\right)-\mathrm{y3}\left(x\right)-\mathrm{y4}\left(x\right)\,\mathrm{diff}\left(\mathrm{y3}\left(x\right)\,x\right)-\mathrm{y5}\left(x\right)\,\mathrm{diff}\left(\mathrm{y4}\left(x\right)\,x\right)-2\mathrm{y1}\left(x\right)-2x\mathrm{y2}\left(x\right)-\mathrm{y5}\left(x\right)\,\mathrm{diff}\left(\mathrm{y5}\left(x\right)\,x\right)-x^2\mathrm{y1}\left(x\right)-2x\mathrm{y3}\left(x\right)-\mathrm{y6}\left(x\right)\,\mathrm{diff}\left(\mathrm{y6}\left(x\right)\,x\right)-x^2\mathrm{y2}\left(x\right)+2\mathrm{y3}\left(x\right)\right]\:$

$\mathrm{vars}≔\left[\mathrm{y1}\left(x\right)\,\mathrm{y2}\left(x\right)\,\mathrm{y3}\left(x\right)\,\mathrm{y4}\left(x\right)\,\mathrm{y5}\left(x\right)\,\mathrm{y6}\left(x\right)\right]\:$

$\mathrm{HomogeneousSystem}\left('\mathrm{homo}'\,\mathrm{sys}\,\mathrm{vars}\right)$

$\left[\frac{\ⅆ}{\ⅆx}\mathrm{y1}\left(x\right)-\mathrm{y2}\left(x\right)\,\frac{\ⅆ}{\ⅆx}\mathrm{y2}\left(x\right)-\mathrm{y3}\left(x\right)-\mathrm{y4}\left(x\right)\,\frac{\ⅆ}{\ⅆx}\mathrm{y3}\left(x\right)-\mathrm{y5}\left(x\right)\,\frac{\ⅆ}{\ⅆx}\mathrm{y4}\left(x\right)-2\mathrm{y1}\left(x\right)-2x\mathrm{y2}\left(x\right)-\mathrm{y5}\left(x\right)\,\frac{\ⅆ}{\ⅆx}\mathrm{y5}\left(x\right)-x^2\mathrm{y1}\left(x\right)-2x\mathrm{y3}\left(x\right)-\mathrm{y6}\left(x\right)\,\frac{\ⅆ}{\ⅆx}\mathrm{y6}\left(x\right)-x^2\mathrm{y2}\left(x\right)+2\mathrm{y3}\left(x\right)\right],\left[\mathrm{y1}\left(x\right)\,\mathrm{y2}\left(x\right)\,\mathrm{y3}\left(x\right)\,\mathrm{y4}\left(x\right)\,\mathrm{y5}\left(x\right)\,\mathrm{y6}\left(x\right)\right]$

$\mathrm{homo}$

$\mathrm{true}$

$\mathrm{sys}≔\left[\mathrm{diff}\left(\mathrm{y1}\left(x\right)\,x\right)\left(-1+x^2\right)+\left(6+2x\right)\mathrm{y1}\left(x\right)+6\mathrm{y3}\left(x\right)-6\mathrm{y4}\left(x\right)=0\,\mathrm{diff}\left(\mathrm{y2}\left(x\right)\,x\right)\cdot 2x{\left(1+x^2\right)}^2-\left(4+x\right)\mathrm{y2}\left(x\right){\left(1+x^2\right)}^2+4\mathrm{y3}\left(x\right){\left(1+x^2\right)}^2=-4x^2-\left(4+x\right)\left(1+x^2\right)\,\mathrm{diff}\left(\mathrm{y3}\left(x\right)\,x\right)\left(1+x^2\right)\cdot 2x-\left(4+3x\right)\left(1+x^2\right)\mathrm{y2}\left(x\right)+4\left(1+x^2\right)\mathrm{y3}\left(x\right)=-\left(4+3x\right)\,\mathrm{diff}\left(\mathrm{y4}\left(x\right)\,x\right)\left(1+x^2\right)\left(x^2-1\right)\cdot 2x-\left(1+x^2\right)\cdot 2x\left(-7+x^2-2x\right)\mathrm{y1}\left(x\right)-\mathrm{y2}\left(x\right)\left(4+3x\right)\left(1+x^2\right)\left(x^2-1\right)+\mathrm{y3}\left(x\right)\left(6x+2x^2-2\right)\left(1+x^2\right)\cdot 2-\mathrm{y4}\left(x\right)\cdot 6\left(1+x^2\right)\cdot 2x=-\left(4+3x\right)\left(x^2-1\right)\right]$

$\mathrm{sys}≔\left[\left(\frac{\ⅆ}{\ⅆx}\mathrm{y1}\left(x\right)\right)\left(x^2-1\right)+\left(6+2x\right)\mathrm{y1}\left(x\right)+6\mathrm{y3}\left(x\right)-6\mathrm{y4}\left(x\right)=0\,2\left(\frac{\ⅆ}{\ⅆx}\mathrm{y2}\left(x\right)\right)x{\left(x^2+1\right)}^2-\left(4+x\right)\mathrm{y2}\left(x\right){\left(x^2+1\right)}^2+4\mathrm{y3}\left(x\right){\left(x^2+1\right)}^2=-4x^2-\left(4+x\right)\left(x^2+1\right)\,2\left(\frac{\ⅆ}{\ⅆx}\mathrm{y3}\left(x\right)\right)\left(x^2+1\right)x-\left(4+3x\right)\left(x^2+1\right)\mathrm{y2}\left(x\right)+4\left(x^2+1\right)\mathrm{y3}\left(x\right)=-4-3x\,2\left(\frac{\ⅆ}{\ⅆx}\mathrm{y4}\left(x\right)\right)\left(x^2+1\right)\left(x^2-1\right)x-2\left(x^2+1\right)x\left(x^2-2x-7\right)\mathrm{y1}\left(x\right)-\mathrm{y2}\left(x\right)\left(4+3x\right)\left(x^2+1\right)\left(x^2-1\right)+2\mathrm{y3}\left(x\right)\left(2x^2+6x-2\right)\left(x^2+1\right)-12\mathrm{y4}\left(x\right)\left(x^2+1\right)x=-\left(4+3x\right)\left(x^2-1\right)\right]$

$\mathrm{vars}≔\left[\mathrm{y1}\left(x\right)\,\mathrm{y2}\left(x\right)\,\mathrm{y3}\left(x\right)\,\mathrm{y4}\left(x\right)\right]\:$

$\mathrm{HomogeneousSystem}\left('\mathrm{homo}'\,\mathrm{sys}\,\mathrm{vars}\right)$

$\left[\left(\frac{\ⅆ}{\ⅆx}\mathrm{y1}\left(x\right)\right)x^2+2\mathrm{y1}\left(x\right)x+6\mathrm{y3}\left(x\right)-6\mathrm{y4}\left(x\right)+6\mathrm{y1}\left(x\right)-\frac{\ⅆ}{\ⅆx}\mathrm{y1}\left(x\right)\,2\left(\frac{\ⅆ}{\ⅆx}\mathrm{y2}\left(x\right)\right)x^5-\mathrm{y2}\left(x\right)x^5+4\mathrm{y3}\left(x\right)x^4-4\mathrm{y2}\left(x\right)x^4+4\left(\frac{\ⅆ}{\ⅆx}\mathrm{y2}\left(x\right)\right)x^3-2\mathrm{y2}\left(x\right)x^3+8\mathrm{y3}\left(x\right)x^2-8x^2\mathrm{y2}\left(x\right)+2\left(\frac{\ⅆ}{\ⅆx}\mathrm{y2}\left(x\right)\right)x-x\mathrm{y2}\left(x\right)+4\mathrm{y3}\left(x\right)-4\mathrm{y2}\left(x\right)+\left(x^3+8x^2+x+4\right)\mathrm{\_yc}\left(x\right)\,2\left(\frac{\ⅆ}{\ⅆx}\mathrm{y3}\left(x\right)\right)x^3-3\mathrm{y2}\left(x\right)x^3+4\mathrm{y3}\left(x\right)x^2-4x^2\mathrm{y2}\left(x\right)+2\left(\frac{\ⅆ}{\ⅆx}\mathrm{y3}\left(x\right)\right)x-3x\mathrm{y2}\left(x\right)+4\mathrm{y3}\left(x\right)-4\mathrm{y2}\left(x\right)+\left(4+3x\right)\mathrm{\_yc}\left(x\right)\,2\left(\frac{\ⅆ}{\ⅆx}\mathrm{y4}\left(x\right)\right)x^5-2\mathrm{y1}\left(x\right)x^5-3\mathrm{y2}\left(x\right)x^5+4\mathrm{y3}\left(x\right)x^4+4\mathrm{y1}\left(x\right)x^4-4\mathrm{y2}\left(x\right)x^4+12\mathrm{y3}\left(x\right)x^3-12\mathrm{y4}\left(x\right)x^3+12\mathrm{y1}\left(x\right)x^3+4x^2\mathrm{y1}\left(x\right)+12x\mathrm{y3}\left(x\right)-12\mathrm{y4}\left(x\right)x-2\left(\frac{\ⅆ}{\ⅆx}\mathrm{y4}\left(x\right)\right)x+14\mathrm{y1}\left(x\right)x+3x\mathrm{y2}\left(x\right)-4\mathrm{y3}\left(x\right)+4\mathrm{y2}\left(x\right)+\left(3x^3+4x^2-3x-4\right)\mathrm{\_yc}\left(x\right)\,\frac{\ⅆ}{\ⅆx}\mathrm{\_yc}\left(x\right)\right],\left[\mathrm{y1}\left(x\right)\,\mathrm{y2}\left(x\right)\,\mathrm{y3}\left(x\right)\,\mathrm{y4}\left(x\right)\,\mathrm{\_yc}\left(x\right)\right]$

$\mathrm{homo}$

$\mathrm{false}$