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DEtools

 reduceOrder
 apply the method of reduction of order to an ODE

 Calling Sequence reduceOrder(des, dvar, partsol, solutionForm)

Parameters

 des - ordinary differential equation, or its list form dvar - the dependent variable for an equation partsol - partial solution, or list of partial solutions solutionForm - flag to indicate the DE should be solved explicitly

Description

 • Important: The DEtools[reduceOrder] command has been deprecated. Use the superseding command DEtools[reduce_order] instead.
 • This routine is used to either a) return an ODE of reduced order or b) solve the ODE explicitly by the method of reduction of order, given a partial (particular) solution of the ODE.  Without the optional flag basis, a reduced ODE is returned.  If basis appears as the fifth argument, then a list containing the basis of the solution is returned.  Note that a solution basis may contain DESol data structures.
 • des may be input as an explicit ODE, as a list of coefficients (in the case of the ODE being homogeneous), or in the form returned by convertAlg for the non-homogeneous case.
 • partsol may be a single partial solution, or a list of partial solutions.  Note that it is assumed all given partial solutions are correct and valid.  When a reduced ODE is to be returned, the order of the resulting ODE will be equal to the order of the original less the number of partial solutions given.
 • The command with(DEtools,reduceOrder) allows the use of the abbreviated form of this command.

Examples

Important: The DEtools[reduceOrder] command has been deprecated. Use the superseding command DEtools[reduce_order] instead.

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $\mathrm{de}≔\mathrm{diff}\left(y\left(x\right),\mathrm{}\left(x,3\right)\right)-6\mathrm{diff}\left(y\left(x\right),\mathrm{}\left(x,2\right)\right)+11\mathrm{diff}\left(y\left(x\right),x\right)-6y\left(x\right)$
 ${\mathrm{de}}{≔}\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}{6}{}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{11}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}{6}{}{y}{}\left({x}\right)$ (1)
 > $\mathrm{sol}≔\mathrm{exp}\left(x\right)$
 ${\mathrm{sol}}{≔}{{ⅇ}}^{{x}}$ (2)
 > $\mathrm{reduceOrder}\left(\mathrm{de},y\left(x\right),\mathrm{sol}\right)$
 $\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}{3}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{2}{}{y}{}\left({x}\right)$ (3)
 > $\mathrm{reduceOrder}\left(\mathrm{de},y\left(x\right),\mathrm{sol},\mathrm{basis}\right)$
 $\left[{{ⅇ}}^{{x}}{,}{{ⅇ}}^{{2}{}{x}}{,}\frac{{{ⅇ}}^{{3}{}{x}}}{{2}}\right]$ (4)
 > $\mathrm{de2}≔\left[24,-50,35,-10,1\right]$
 ${\mathrm{de2}}{≔}\left[{24}{,}{-50}{,}{35}{,}{-10}{,}{1}\right]$ (5)
 > $\mathrm{sol1}≔\mathrm{exp}\left(x\right)$
 ${\mathrm{sol1}}{≔}{{ⅇ}}^{{x}}$ (6)
 > $\mathrm{sol2}≔\mathrm{exp}\left(2x\right)$
 ${\mathrm{sol2}}{≔}{{ⅇ}}^{{2}{}{x}}$ (7)
 > $\mathrm{reduceOrder}\left(\mathrm{de2},y\left(x\right),\mathrm{sol1}\right)$
 $\left[{-6}{,}{11}{,}{-6}{,}{1}\right]$ (8)
 > $\mathrm{reduceOrder}\left(\mathrm{de2},y\left(x\right),\mathrm{sol2}\right)$
 $\left[{2}{,}{-1}{,}{-2}{,}{1}\right]$ (9)
 > $\mathrm{reduceOrder}\left(\mathrm{de2},y\left(x\right),\left[\mathrm{sol1},\mathrm{sol2}\right]\right)$
 $\left[{2}{,}{-3}{,}{1}\right]$ (10)
 > $\mathrm{reduceOrder}\left(\mathrm{de2},y\left(x\right),\left[\mathrm{sol1},\mathrm{sol2}\right],\mathrm{basis}\right)$
 $\left[{{ⅇ}}^{{x}}{,}{{ⅇ}}^{{2}{}{x}}{,}\frac{{{ⅇ}}^{{3}{}{x}}}{{2}}{,}\frac{{{ⅇ}}^{{4}{}{x}}}{{6}}\right]$ (11)
 > $\mathrm{de3}≔\left({x}^{9}+{x}^{3}\right)\mathrm{diff}\left(y\left(x\right),\mathrm{}\left(x,3\right)\right)+18{x}^{8}\mathrm{diff}\left(y\left(x\right),\mathrm{}\left(x,2\right)\right)-90x\mathrm{diff}\left(y\left(x\right),x\right)-30\left(11{x}^{6}-3\right)y\left(x\right)$
 ${\mathrm{de3}}{≔}\left({{x}}^{{9}}{+}{{x}}^{{3}}\right){}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{18}{}{{x}}^{{8}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{90}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{30}{}\left({11}{}{{x}}^{{6}}{-}{3}\right){}{y}{}\left({x}\right)$ (12)
 > $\mathrm{sol}≔\frac{x}{{x}^{6}+1}$
 ${\mathrm{sol}}{≔}\frac{{x}}{{{x}}^{{6}}{+}{1}}$ (13)
 > $\mathrm{reduceOrder}\left(\mathrm{de3},y\left(x\right),\mathrm{sol}\right)$
 ${{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{3}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{90}{}{y}{}\left({x}\right)$ (14)
 > $\mathrm{reduceOrder}\left(\mathrm{de3},y\left(x\right),\mathrm{sol},\mathrm{basis}\right)$
 $\left[\frac{{x}}{{{x}}^{{6}}{+}{1}}{,}{-}\frac{{x}{}\sqrt{{91}}{}{{x}}^{{-}\sqrt{{91}}}}{{91}{}\left({{x}}^{{6}}{+}{1}\right)}{,}\frac{{x}{}\sqrt{{91}}{}{{x}}^{\sqrt{{91}}}}{{91}{}\left({{x}}^{{6}}{+}{1}\right)}\right]$ (15)