AreSameSolution - Maple Help

QDifferenceEquations

 IsSolution
 test the solution of a linear q-difference equation
 AreSameSolution
 test if solutions of a linear q-difference equation are equivalent

 Calling Sequence IsSolution(sol, eq, var, ['certificates']) AreSameSolution(sol, sol1, x)

Parameters

 sol - solution to be checked, or a certificate (set or list of certificates) for q-hypergeometric solution eq - q-difference equation or list of such equations (for the system case) var - function variable to solve for, such as $y\left(x\right)$, or a list of such function variables (for the system case) 'certificates' - (optional) if given sol then treated as certificate for q-hypergeometric solution sol1 - solution to be compared with x - independent variable in the solutions given

Description

 • The IsSolution(sol,eq,var) calling sequence returns true if eq is a linear q-difference equation with polynomial coefficients (or a system of such equations) and sol is its solution. Otherwise, false is returned for q-hypergeometric solution (that is, if 'certificates' option is specified) and the degree of the result of substitution is returned for other types of solutions to be checked.
 The IsSolution command substitutes sol for the function variables and checks the result.
 • The AreSameSolution(sol, sol1, x) calling sequence returns true if the solutions sol and sol1 are equivalent. Otherwise, false is returned.
 The command checks that the numbers of linear independent solutions in sol and sol1 are the same, as well as the degrees in x.

Examples

 > $\mathrm{with}\left(\mathrm{QDifferenceEquations}\right):$
 > $\mathrm{eq}≔\left(1-{q}^{10}-\left(q-{q}^{10}\right)x\right)y\left({q}^{2}x\right)-\left(1-{q}^{20}-\left({q}^{2}-{q}^{20}\right)x\right)y\left(qx\right)+{q}^{10}\left(1-{q}^{10}-\left({q}^{2}-{q}^{11}\right)x\right)y\left(x\right)=\left({q}^{21}-{q}^{20}-{q}^{12}+{q}^{10}+{q}^{2}-q\right)x$
 ${\mathrm{eq}}{≔}\left({1}{-}{{q}}^{{10}}{-}\left({-}{{q}}^{{10}}{+}{q}\right){}{x}\right){}{y}{}\left({{q}}^{{2}}{}{x}\right){-}\left({1}{-}{{q}}^{{20}}{-}\left({-}{{q}}^{{20}}{+}{{q}}^{{2}}\right){}{x}\right){}{y}{}\left({q}{}{x}\right){+}{{q}}^{{10}}{}\left({1}{-}{{q}}^{{10}}{-}\left({-}{{q}}^{{11}}{+}{{q}}^{{2}}\right){}{x}\right){}{y}{}\left({x}\right){=}\left({{q}}^{{21}}{-}{{q}}^{{20}}{-}{{q}}^{{12}}{+}{{q}}^{{10}}{+}{{q}}^{{2}}{-}{q}\right){}{x}$ (1)
 > $\mathrm{sol1}≔\mathrm{PolynomialSolution}\left(\mathrm{eq},y\left(x\right),\varnothing ,\mathrm{output}=\mathrm{basis}\left[\mathrm{_K}\right]\right)$
 ${\mathrm{sol1}}{≔}{{\mathrm{_K}}}_{{1}}{}{{x}}^{{10}}{+}{{\mathrm{_K}}}_{{2}}{}{x}{-}{{\mathrm{_K}}}_{{2}}{+}{1}$ (2)
 > $\mathrm{IsSolution}\left(\mathrm{sol1},\mathrm{eq},y\left(x\right)\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{sol2}≔\mathrm{sol1}-{x}^{5}$
 ${\mathrm{sol2}}{≔}{{\mathrm{_K}}}_{{1}}{}{{x}}^{{10}}{-}{{x}}^{{5}}{+}{{\mathrm{_K}}}_{{2}}{}{x}{-}{{\mathrm{_K}}}_{{2}}{+}{1}$ (4)
 > $\mathrm{IsSolution}\left(\mathrm{sol2},\mathrm{eq},y\left(x\right)\right)$
 ${6}$ (5)
 > $\mathrm{sol3}≔\mathrm{sol1}-{x}^{10}$
 ${\mathrm{sol3}}{≔}{{\mathrm{_K}}}_{{1}}{}{{x}}^{{10}}{-}{{x}}^{{10}}{+}{{\mathrm{_K}}}_{{2}}{}{x}{-}{{\mathrm{_K}}}_{{2}}{+}{1}$ (6)
 > $\mathrm{IsSolution}\left(\mathrm{sol3},\mathrm{eq},y\left(x\right)\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{AreSameSolution}\left(\mathrm{sol1},\mathrm{sol3},x\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{sol4}≔\mathrm{sol1}-\mathrm{_K}\left[1\right]{x}^{10}$
 ${\mathrm{sol4}}{≔}{{\mathrm{_K}}}_{{2}}{}{x}{-}{{\mathrm{_K}}}_{{2}}{+}{1}$ (9)
 > $\mathrm{IsSolution}\left(\mathrm{sol4},\mathrm{eq},y\left(x\right)\right)$
 ${\mathrm{true}}$ (10)
 > $\mathrm{AreSameSolution}\left(\mathrm{sol1},\mathrm{sol4},x\right)$
 ${\mathrm{false}}$ (11)