The ByPerturbation(ODE, y(x)) command finds the solution of a second order ODE by the perturbation method.

Use the option output=steps to make this command return an annotated step-by-step solution.  Further control over the format and display of the step-by-step solution is available using the options described in Student:-Basics:-OutputStepsRecord.  The options supported by that command can be passed to this one.

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

$\mathrm{ode1}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+y\left(x\right)-\sin \left(\mathrm{\epsilon }y\left(x\right)\right)=0$

$\mathrm{ode1}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+y\left(x\right)-\sin \left(\mathrm{\epsilon }y\left(x\right)\right)=0$

$\mathrm{ic1}≔\left\{\mathrm{eval}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)\,x=0\right)=1\,y\left(0\right)=0\right\}$

$\mathrm{ic1}≔\left\{\genfrac{}{}{0ex}{}{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}{\phantom{\left\{x=0\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}}{\left\{x=0\right\}}=1\,y\left(0\right)=0\right\}$

$\mathrm{ByPerturbation}\left(\mathrm{ode1}\,\mathrm{ic1}\,y\left(x\right)\,\mathrm{\epsilon }\,3\right)$

$\left[y\left(\mathrm{\tau }\right)=\sin \left(\mathrm{\tau }\right)+\frac{{\sin \left(\mathrm{\tau }\right)}^3{\mathrm{\epsilon }}^3}{48}+\mathrm{O}\left({\mathrm{\epsilon }}^4\right)\,\mathrm{\tau }=\left(-\frac{{\mathrm{\epsilon }}^2}{8}-\frac{\mathrm{\epsilon }}{2}+1+\mathrm{O}\left({\mathrm{\epsilon }}^4\right)\right)x\right]$

$\mathrm{ode2}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+4y\left(x\right)+2\mathrm{diff}\left(y\left(x\right)\,x\right)+\mathrm{\epsilon }y\left(x\right)+\cos \left(\mathrm{\epsilon }y\left(x\right)\right)=0$

$\mathrm{ode2}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+4y\left(x\right)+2\frac{\ⅆ}{\ⅆx}y\left(x\right)+\mathrm{\epsilon }y\left(x\right)+\cos \left(\mathrm{\epsilon }y\left(x\right)\right)=0$

$\mathrm{ic2}≔\left\{\mathrm{eval}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)\,x=0\right)=0\,y\left(0\right)=-\frac{1}{4}\right\}$

$\mathrm{ic2}≔\left\{\genfrac{}{}{0ex}{}{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}{\phantom{\left\{x=0\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}}{\left\{x=0\right\}}=0\,y\left(0\right)=-\frac{1}{4}\right\}$

$\mathrm{ByPerturbation}\left(\mathrm{ode2}\,\mathrm{ic2}\,y\left(x\right)\,\mathrm{\epsilon }\,1\right)$

$y\left(x\right)=-\frac{1}{4}+\left(-\frac{{\ⅇ}^{-x}\cos \left(\sqrt{3}x\right)}{16}-\frac{{\ⅇ}^{-x}\sin \left(\sqrt{3}x\right)\sqrt{3}}{48}+\frac{1}{16}\right)\mathrm{\epsilon }$

$\mathrm{ode3}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+y\left(x\right)+\mathrm{\epsilon }{y\left(x\right)}^3=0$

$\mathrm{ode3}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+y\left(x\right)+\mathrm{\epsilon }{y\left(x\right)}^3=0$

$\mathrm{ic3}≔\left\{\mathrm{eval}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)\,x=0\right)=0\,y\left(0\right)=1\right\}$

$\mathrm{ic3}≔\left\{\genfrac{}{}{0ex}{}{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}{\phantom{\left\{x=0\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}}{\left\{x=0\right\}}=0\,y\left(0\right)=1\right\}$

$\mathrm{ByPerturbation}\left(\mathrm{ode3}\,\mathrm{ic3}\,y\left(x\right)\,\mathrm{\epsilon }\,2\right)$

$\left[y\left(\mathrm{\tau }\right)=\cos \left(\mathrm{\tau }\right)-\frac{\cos \left(\mathrm{\tau }\right){\sin \left(\mathrm{\tau }\right)}^2\mathrm{\epsilon }}{8}+\frac{{\sin \left(\mathrm{\tau }\right)}^2\left(-{\cos \left(\mathrm{\tau }\right)}^3+\frac{25\cos \left(\mathrm{\tau }\right)}{4}\right){\mathrm{\epsilon }}^2}{64}+\mathrm{O}\left({\mathrm{\epsilon }}^3\right)\,\mathrm{\tau }=\left(-\frac{21{\mathrm{\epsilon }}^2}{256}+\frac{3\mathrm{\epsilon }}{8}+1+\mathrm{O}\left({\mathrm{\epsilon }}^3\right)\right)x\right]$

$\mathrm{ode4}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+\left(-\mathrm{\epsilon }x+1\right)y\left(x\right)=0$

$\mathrm{ode4}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+\left(-\mathrm{\epsilon }x+1\right)y\left(x\right)=0$

$\mathrm{ic4}≔\left\{\mathrm{eval}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)\,x=0\right)=0\,y\left(0\right)=1\right\}$

$\mathrm{ic4}≔\left\{\genfrac{}{}{0ex}{}{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}{\phantom{\left\{x=0\right\}}}|\genfrac{}{}{0ex}{}{\phantom{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}}{\left\{x=0\right\}}=0\,y\left(0\right)=1\right\}$

$\mathrm{ByPerturbation}\left(\mathrm{ode4}\,\mathrm{ic4}\,y\left(x\right)\,\mathrm{\epsilon }\,2\right)$

$y\left(x\right)=\cos \left(x\right)+\left(\frac{\cos \left(x\right)x}{4}+\frac{\sin \left(x\right)x^2}{4}-\frac{\sin \left(x\right)}{4}\right)\mathrm{\epsilon }-\frac{x\left(\left(x^3-7x\right)\cos \left(x\right)+\sin \left(x\right)\left(-\frac{10x^2}{3}+7\right)\right){\mathrm{\epsilon }}^2}{32}$

The Student[ODEs][Solve][ByPerturbation] command was introduced in Maple 2021.

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

The Student[ODEs][Solve][ByPerturbation] command was updated in Maple 2022.

The output option was introduced in Maple 2022.

For more information on Maple 2022 changes, see Updates in Maple 2022.