The LinearParticularSolution(ODE, homsol) command reduces the order of a second order linear ODE using a solution of the corresponding homogeneous ODE.

The second argument must be a solution of the corresponding homogeneous ODE of the form y(x) = homsol, where homsol 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 the 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 is returned.

If the option solve is given and, furthermore, the option output=basis is given, then as above an attempt is made to find the general solution to the original ODE, and the answer is returned in the form of a sequence. The first element of the sequence is a particular solution of the ODE, and the second element is a basis of solutions for the homogeneous solution.

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

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

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

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

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

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

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

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

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

$\mathrm{reduced\_sol}≔u\left(t\right)=\left(\left(t-1\right){\ⅇ}^t+\mathrm{\_C1}\right){\ⅇ}^{-2t}$

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

$\mathrm{new\_ode}≔\frac{\frac{\ⅆ}{\ⅆx}y\left(x\right)}{{\ⅇ}^x}-\frac{y\left(x\right)}{{\ⅇ}^x}=\left(\left(x-1\right){\ⅇ}^x+\mathrm{\_C1}\right){\ⅇ}^{-2x}$

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

$\mathrm{gensol1}≔y\left(x\right)=\mathrm{\_C2}{\ⅇ}^x-x-\frac{\mathrm{c\_\_1}{\ⅇ}^{-x}}{2}$

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

$\mathrm{gensol}≔y\left(x\right)={\ⅇ}^x\left(-\frac{{\ⅇ}^{-2x}\mathrm{c\_\_1}}{2}-{\ⅇ}^{-x}x+\mathrm{\_C2}\right)$

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

$y\left(x\right)=\mathrm{\_C2}{\ⅇ}^x-x-\frac{\mathrm{c\_\_1}{\ⅇ}^{-x}}{2}$

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

$\mathrm{psol},\mathrm{basis}≔-x,\left[{\ⅇ}^{-x}\,{\ⅇ}^x\right]$

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

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

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

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

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

$2$

$\mathrm{LinearParticularSolution}\left(\mathrm{ode}\,y\left(x\right)=\exp \left(-x\right)\,v\left(s\right)\right)$

$\frac{\ⅆ}{\ⅆs}v\left(s\right)=\frac{2v\left(s\right){\ⅇ}^{-s}+s}{{\ⅇ}^{-s}},\left\{s=x\,v\left(s\right)=\frac{\ⅆ}{\ⅆx}\left(\frac{y\left(x\right)}{{\ⅇ}^{-x}}\right)\right\}$

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

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