The BySeries(ODE, y(x)) command finds a particular series solution of a linear homogeneous ODE with polynomial coefficients.

Note that the series solution may not represent the complete solution of the given ODE.

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

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

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

$\mathrm{BySeries}\left(\mathrm{ode1}\,y\left(x\right)\right)$

$\left[y\left(x\right)=\sum _{k=0}^{\mathrm{\infty }}{a}_k{x}^k\,a_{k+2}=-\frac{a_k}{k+2}\right]$

$\mathrm{ode3}≔x^2\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+x^2\mathrm{diff}\left(y\left(x\right)\,x\right)+\left(x^3-6\right)y\left(x\right)=0$

$\mathrm{ode3}≔x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+x^2\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(x^3-6\right)y\left(x\right)=0$

$\mathrm{BySeries}\left(\mathrm{ode3}\,y\left(x\right)\right)$

$\left[y\left(x\right)=\sum _{k=0}^{\mathrm{\infty }}{a}_k{x}^{k+3}\,a_{k+3}=-\frac{k{a}_{k+2}+a_k+5a_{k+2}}{\left(k+8\right)\left(k+3\right)}\,a_1=-\frac{a_0}{2}\,a_2=\frac{a_0}{7}\right]$

$\mathrm{ode4}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+\mathrm{diff}\left(y\left(x\right)\,x\right)+x^{2}y\left(x\right)=0$

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

$\mathrm{BySeries}\left(\mathrm{ode4}\,y\left(x\right)\right)$

$\left[y\left(x\right)=\sum _{k=0}^{\mathrm{\infty }}{a}_k{x}^k\,a_{k+4}=-\frac{k{a}_{k+3}+a_k+3a_{k+3}}{k^2+7k+12}\,a_2=-\frac{a_1}{2}\,a_3=\frac{a_1}{6}\right]$

$\mathrm{ode5}≔\mathrm{diff}\left(\left(-x^2+1\right)\mathrm{diff}\left(y\left(x\right)\,x\right)\,x\right)+12y\left(x\right)=0$

$\mathrm{ode5}≔-2x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(-x^2+1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+12y\left(x\right)=0$

$\mathrm{BySeries}\left(\mathrm{ode5}\,y\left(x\right)\right)$

$y\left(x\right)=a_0\left(\frac{3}{2}x-\frac{5}{2}{x}^3\right)$

$\mathrm{ode6}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)=\sin \left(x\right)y\left(x\right)$

$\mathrm{ode6}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\sin \left(x\right)y\left(x\right)$

$\mathrm{BySeries}\left(\mathrm{ode6}\,y\left(x\right)\right)$

Error, (in Student:-ODEs:-SeriesSolve) series solutions are only available for linear homogeneous ODEs with polynomial coefficients

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

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