The LinearParticularSolution(ODE, psol) command reduces the order of a second order linear ODE using a particular solution.

The second argument must be a particular solution of the ODE of the form y(x) = psol, where psol does not depend on y(x).

The third argument, u(t), representing the variable for the reduced ODE, is optional. If it is not given, new independent and dependent variables will be chosen which do not conflict with the existing variables.

The default output is a sequence consisting of the reduced ODE in terms of the new variables, followed by the transformation used to recover the original ODE from the reduced ODE.

If an extra option solve or solve=true is also given, an attempt is made to solve the reduced ODE and return the general solution to the original ODE. If successful, the general solution of the original ODE will be returned.

If the option solve is given and furthermore, the extra option output=basis is given, then as above an attempt will be made to find the general solution to the original ODE, but the answer will be returned in the form of a list of particular solutions forming a basis for that general solution.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\right)\:$

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\left[\mathrm{ReduceOrder}\right]\right)\:$

$\mathrm{ode}≔\left(x-1\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-x\mathrm{diff}\left(y\left(x\right)\,x\right)+y\left(x\right)=0$

$\mathrm{ode}≔\left(x-1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+y\left(x\right)=0$

$\mathrm{psol}≔y\left(x\right)=\exp \left(x\right)$

$\mathrm{psol}≔y\left(x\right)={\ⅇ}^x$

$\mathrm{reduced\_ode},\mathrm{tr}≔\mathrm{LinearParticularSolution}\left(\mathrm{ode}\,\mathrm{psol}\right)$

$\mathrm{reduced\_ode},\mathrm{tr}≔\frac{\ⅆ}{\ⅆt}u\left(t\right)=-\frac{u\left(t\right)\left(t-2\right)}{t-1},\left\{t=x\,u\left(t\right)=\frac{\ⅆ}{\ⅆx}\left(\frac{y\left(x\right)}{{\ⅇ}^x}\right)\right\}$

$\mathrm{reduced\_sol}≔\mathrm{Solve}\left(\mathrm{reduced\_ode}\,u\left(t\right)\right)$

$\mathrm{reduced\_sol}≔u\left(t\right)={\ⅇ}^{-t+\mathrm{\_C1}}\left(t-1\right)$

$\mathrm{new\_ode}≔\mathrm{eval}\left(\mathrm{reduced\_sol}\,\mathrm{tr}\right)$

$\mathrm{new\_ode}≔\frac{\frac{\ⅆ}{\ⅆx}y\left(x\right)}{{\ⅇ}^x}-\frac{y\left(x\right)}{{\ⅇ}^x}={\ⅇ}^{-x+\mathrm{\_C1}}\left(x-1\right)$

$\mathrm{gensol1}≔\mathrm{Solve}\left(\mathrm{new\_ode}\,y\left(x\right)\right)$

$\mathrm{gensol1}≔y\left(x\right)=-{\ⅇ}^x\left(x{\ⅇ}^{-x+\mathrm{\_C1}}-\mathrm{\_C2}\right)$

$\mathrm{gensol}≔\mathrm{LinearParticularSolution}\left(\mathrm{ode}\,\mathrm{psol}\,'\mathrm{solve}'\right)$

$\mathrm{gensol}≔y\left(x\right)={\ⅇ}^x\left(-x{\ⅇ}^{-x+\mathrm{\_C1}}+\mathrm{\_C2}\right)$

$\mathrm{simplify}\left(\mathrm{expand}\left(\mathrm{gensol}\right)\right)$

$y\left(x\right)=-x{\ⅇ}^{\mathrm{\_C1}}+{\ⅇ}^x\mathrm{\_C2}$

$\mathrm{Basis}≔\mathrm{LinearParticularSolution}\left(\mathrm{ode}\,\mathrm{psol}\,'\mathrm{solve}'\,'\mathrm{output}'='\mathrm{basis}'\right)$

$\mathrm{Basis}≔\left[x\,{\ⅇ}^x\right]$

$\mathrm{sol}≔y\left(x\right)=\mathrm{remove}\left(\mathrm{`=`}\,\mathrm{Basis}\,\mathrm{rhs}\left(\mathrm{psol}\right)\right)\left[1\right]$

$\mathrm{sol}≔y\left(x\right)=x$

$W≔\mathrm{VectorCalculus}:-\mathrm{Wronskian}\left(\mathrm{Basis}\,x\right)$

$W≔\left[\begin{array}{cc}x& {\ⅇ}^x\\ 1& {\ⅇ}^x\end{array}\right]$

$\mathrm{simplify}\left(\mathrm{LinearAlgebra}:-\mathrm{Determinant}\left(W\right)\right)$

${\ⅇ}^x\left(x-1\right)$

The Student[ODEs][ReduceOrder][LinearParticularSolution] command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.