For high order ODEs it may happen that dsolve succeeds in reducing the order of the ODE but not in solving the problem to the end. In those cases, the solution is expressed using ODESolStruc. You can obtain a solution for the reduced ODE by using the tools available in DEtools, as a series expansion, or by other means. If a solution to the reduced ODE is obtained, you can build a solution to the original problem using the buildsol command.

The buildsol command is a function of two arguments. The first argument is the structure returned by dsolve as the solution to an ODE (see ODESolStruc). The second argument is a solution found by the user to the reduced ODE present inside the first argument. That solution may be a particular one; buildsol will not check whether the given solution actually solves the reduced ODE.

This function is part of the DEtools package, and so it can be used in the form buildsol(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[buildsol](..).

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

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

$\mathrm{ODE}≔\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)=\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)\left(-1+\frac{\ⅆ}{\ⅆx}y\left(x\right)\right){\ⅇ}^{y\left(x\right)-x}$

$\mathrm{odeadvisor}\left(\mathrm{ODE}\right)$

$\left[\left[\mathrm{\_3rd\_order}\,\mathrm{\_with\_linear\_symmetries}\right]\,\left[\mathrm{\_3rd\_order}\,\mathrm{\_reducible}\,\mathrm{\_mu\_y2}\right]\,\left[\mathrm{\_3rd\_order}\,\mathrm{\_reducible}\,\mathrm{\_mu\_poly\_yn}\right]\right]$

$\mathrm{sol}≔y\left(x\right)=\mathrm{ODESolStruc}\left(\mathrm{\_c}+\mathrm{Int}\left(\exp \left(\mathrm{Int}\left(\mathrm{\_f1}\left(\mathrm{\_c}\right)\,\mathrm{\_c}\right)+\mathrm{\_C1}\right)\,\mathrm{\_c}\right)+\mathrm{\_C2}\,\left[\left\{\mathrm{diff}\left(\mathrm{\_f1}\left(\mathrm{\_c}\right)\,\mathrm{\_c}\right)=2{\mathrm{\_f1}\left(\mathrm{\_c}\right)}^2+\mathrm{\_f1}\left(\mathrm{\_c}\right)\exp \left(\mathrm{\_c}\right)\right\}\,\left\{\mathrm{\_c}=y\left(x\right)-x\,\mathrm{\_f1}\left(\mathrm{\_c}\right)=-\frac{\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)\,x\right)}{{\left(-1+\mathrm{diff}\left(y\left(x\right)\,x\right)\right)}^2}\right\}\,\left\{x=\mathrm{Int}\left(\exp \left(\mathrm{Int}\left(\mathrm{\_f1}\left(\mathrm{\_c}\right)\,\mathrm{\_c}\right)+\mathrm{\_C1}\right)\,\mathrm{\_c}\right)+\mathrm{\_C2}\,y\left(x\right)=\mathrm{\_c}+\mathrm{Int}\left(\exp \left(\mathrm{Int}\left(\mathrm{\_f1}\left(\mathrm{\_c}\right)\,\mathrm{\_c}\right)+\mathrm{\_C1}\right)\,\mathrm{\_c}\right)+\mathrm{\_C2}\right\}\right]\right)$

$\mathrm{sol}≔y\left(x\right)=\left(\mathrm{\_c}+\int {\ⅇ}^{\int \mathrm{\_f1}\left(\mathrm{\_c}\right)\ⅆ\mathrm{\_c}+\mathrm{\_C1}}\ⅆ\mathrm{\_c}+\mathrm{\_C2}\right)\mathbf{where}\left[\left\{\frac{\ⅆ}{\ⅆ\mathrm{\_c}}\mathrm{\_f1}\left(\mathrm{\_c}\right)=2{\mathrm{\_f1}\left(\mathrm{\_c}\right)}^2+\mathrm{\_f1}\left(\mathrm{\_c}\right){\ⅇ}^{\mathrm{\_c}}\right\}\,\left\{\mathrm{\_c}=y\left(x\right)-x\,\mathrm{\_f1}\left(\mathrm{\_c}\right)=-\frac{\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)}{{\left(-1+\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2}\right\}\,\left\{x=\int {\ⅇ}^{\int \mathrm{\_f1}\left(\mathrm{\_c}\right)\ⅆ\mathrm{\_c}+\mathrm{\_C1}}\ⅆ\mathrm{\_c}+\mathrm{\_C2}\,y\left(x\right)=\mathrm{\_c}+\int {\ⅇ}^{\int \mathrm{\_f1}\left(\mathrm{\_c}\right)\ⅆ\mathrm{\_c}+\mathrm{\_C1}}\ⅆ\mathrm{\_c}+\mathrm{\_C2}\right\}\right]$

$\mathrm{odetest}\left(\mathrm{sol}\,\mathrm{ODE}\right)$

$0$

$\mathrm{reduced\_ODE}≔\mathrm{op}\left(\left[2\,2\,1\,1\right]\,\mathrm{sol}\right)$

$\mathrm{reduced\_ODE}≔\frac{\ⅆ}{\ⅆ\mathrm{\_c}}\mathrm{\_f1}\left(\mathrm{\_c}\right)=2{\mathrm{\_f1}\left(\mathrm{\_c}\right)}^2+\mathrm{\_f1}\left(\mathrm{\_c}\right){\ⅇ}^{\mathrm{\_c}}$

$\mathrm{odeadvisor}\left(\mathrm{reduced\_ODE}\right)$

$\left[\mathrm{\_Bernoulli}\right]$

$\mathrm{sol\_red\_1}≔\mathrm{\_f1}\left(\mathrm{\_c}\right)=0$

$\mathrm{sol\_red\_1}≔\mathrm{\_f1}\left(\mathrm{\_c}\right)=0$

$\mathrm{particular\_sol}≔\mathrm{buildsol}\left(\mathrm{sol}\,\mathrm{sol\_red\_1}\right)$

$\mathrm{particular\_sol}≔y\left(x\right)=-\frac{-{\ⅇ}^{\mathrm{c\_\_1}}x+\mathrm{c\_\_2}-x}{{\ⅇ}^{\mathrm{c\_\_1}}}$

$\mathrm{odetest}\left(\mathrm{particular\_sol}\,\mathrm{ODE}\right)$

$0$

$\mathrm{sol\_red\_2}≔\mathrm{dsolve}\left(\mathrm{reduced\_ODE}\right)$

$\mathrm{sol\_red\_2}≔\mathrm{\_f1}\left(\mathrm{\_c}\right)=\frac{{\ⅇ}^{{\ⅇ}^{\mathrm{\_c}}}}{2{\mathrm{Ei}}_1\left(-{\ⅇ}^{\mathrm{\_c}}\right)+\mathrm{c\_\_1}}$

$\mathrm{general\_sol}≔\mathrm{buildsol}\left(\mathrm{sol}\,\mathrm{sol\_red\_2}\right)$

$\mathrm{general\_sol}≔y\left(x\right)=x+\mathrm{RootOf}\left(-x+\sqrt{2}\left({\int }_{\phantom{\mathrm{`\; `}}}^{\mathrm{\_Z}}\frac{{\ⅇ}^{\mathrm{c\_\_1}}}{\sqrt{2{\mathrm{Ei}}_1\left(-{\ⅇ}^{\mathrm{\_b}}\right)+\mathrm{c\_\_3}}}\ⅆ\mathrm{\_b}\right)+\mathrm{c\_\_2}\right)$

$\mathrm{odetest}\left(\mathrm{general\_sol}\,\mathrm{ODE}\right)$

$0$