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.

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

$\mathrm{de}≔\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,3\right)\right)-6\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)+11\mathrm{diff}\left(y\left(x\right)\,x\right)-6y\left(x\right)$

$\mathrm{de}≔\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)-6\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+11\frac{\ⅆ}{\ⅆx}y\left(x\right)-6y\left(x\right)$

$\mathrm{sol}≔\exp \left(x\right)$

$\mathrm{sol}≔{\ⅇ}^x$

$\mathrm{reduceOrder}\left(\mathrm{de}\,y\left(x\right)\,\mathrm{sol}\right)$

$\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)-3\frac{\ⅆ}{\ⅆx}y\left(x\right)+2y\left(x\right)$

$\mathrm{reduceOrder}\left(\mathrm{de}\,y\left(x\right)\,\mathrm{sol}\,\mathrm{basis}\right)$

$\left[{\ⅇ}^x\,{\ⅇ}^{2x}\,\frac{{\ⅇ}^{3x}}{2}\right]$

$\mathrm{de2}≔\left[24\,-50\,35\,-10\,1\right]$

$\mathrm{de2}≔\left[24\,\mathrm{-50}\,35\,\mathrm{-10}\,1\right]$

$\mathrm{sol1}≔\exp \left(x\right)$

$\mathrm{sol1}≔{\ⅇ}^x$

$\mathrm{sol2}≔\exp \left(2x\right)$

$\mathrm{sol2}≔{\ⅇ}^{2x}$

$\mathrm{reduceOrder}\left(\mathrm{de2}\,y\left(x\right)\,\mathrm{sol1}\right)$

$\left[\mathrm{-6}\,11\,\mathrm{-6}\,1\right]$

$\mathrm{reduceOrder}\left(\mathrm{de2}\,y\left(x\right)\,\mathrm{sol2}\right)$

$\left[2\,\mathrm{-1}\,\mathrm{-2}\,1\right]$

$\mathrm{reduceOrder}\left(\mathrm{de2}\,y\left(x\right)\,\left[\mathrm{sol1}\,\mathrm{sol2}\right]\right)$

$\left[2\,\mathrm{-3}\,1\right]$

$\mathrm{reduceOrder}\left(\mathrm{de2}\,y\left(x\right)\,\left[\mathrm{sol1}\,\mathrm{sol2}\right]\,\mathrm{basis}\right)$

$\left[{\ⅇ}^x\,{\ⅇ}^{2x}\,\frac{{\ⅇ}^{3x}}{2}\,\frac{{\ⅇ}^{4x}}{6}\right]$

$\mathrm{de3}≔\left(x^9+x^3\right)\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,3\right)\right)+18x^8\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)-90x\mathrm{diff}\left(y\left(x\right)\,x\right)-30\left(11x^6-3\right)y\left(x\right)$

$\mathrm{de3}≔\left(x^9+x^3\right)\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)+18x^8\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-90x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-30\left(11x^6-3\right)y\left(x\right)$

$\mathrm{sol}≔\frac{x}{x^6+1}$

$\mathrm{sol}≔\frac{x}{x^6+1}$

$\mathrm{reduceOrder}\left(\mathrm{de3}\,y\left(x\right)\,\mathrm{sol}\right)$

$x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+3x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-90y\left(x\right)$

$\mathrm{reduceOrder}\left(\mathrm{de3}\,y\left(x\right)\,\mathrm{sol}\,\mathrm{basis}\right)$

$\left[\frac{x}{x^6+1}\,-\frac{x{x}^{-\sqrt{91}}\sqrt{91}}{91\left(x^6+1\right)}\,\frac{x{x}^{\sqrt{91}}\sqrt{91}}{91\left(x^6+1\right)}\right]$