The series_by_leastsquare command determines from the given formal power series a series whose coefficients best satisfy the given linear conditions in the least squares sense and returns the result in form of an FPSstruct data structure.

This command can be used in conjunction with Slode[FPseries] or Slode[FTseries] to construct a least-squares best fit power series solution for a linear ODE with respect to a system of linear constraint equations for some coefficients of the series solution. After constructing a formal series solution FS via Slode[FTseries] or Slode[FPseries], use the series_by_leastsquare command function with the result FS and the linear system as arguments.

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

$\mathrm{ode1}≔\left(x-1\right)y\left(x\right)+\left(x^2-1\right)\mathrm{diff}\left(y\left(x\right)\,x\right)=0$

$\mathrm{ode1}≔\left(x-1\right)y\left(x\right)+\left(x^2-1\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)=0$

$\mathrm{sys1}≔\left\{v\left(8\right)\cdot 50=v\left(10\right)\,v\left(10\right)-v\left(3\right)\cdot 8+v\left(8\right)\cdot 2=64\,v\left(3\right)+v\left(8\right)\cdot 4=13\right\}$

$\mathrm{sys1}≔\left\{50v\left(8\right)=v\left(10\right)\,v\left(3\right)+4v\left(8\right)=13\,v\left(10\right)-8v\left(3\right)+2v\left(8\right)=64\right\}$

$\mathrm{fps1}≔\mathrm{FPseries}\left(\mathrm{ode1}\,y\left(x\right)\,v\left(n\right)\right)\:$

$\mathrm{series\_by\_leastsquare}\left(\mathrm{fps1}\,\mathrm{sys1}\right)$

$\mathrm{FPSstruct}\left(\frac{2120213}{392471}-\frac{2120213x}{392471}+\frac{2120213x^2}{392471}-\frac{2120213x^3}{392471}+\frac{2120213x^4}{392471}-\frac{2120213x^5}{392471}+\frac{2120213x^6}{392471}-\frac{2120213x^7}{392471}+\frac{2120213x^8}{392471}-\frac{2120213x^9}{392471}+\frac{2120213x^{10}}{392471}+\left(\sum _{n=11}^{\mathrm{\infty }}v\left(n\right)x^n\right)\,nv\left(n-1\right)+nv\left(n\right)\right)$

$\mathrm{ode2}≔\left(x-1\right)y\left(x\right)+\left(x^2-1\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)=0$

$\mathrm{ode2}≔\left(x-1\right)y\left(x\right)+\left(x^2-1\right)\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)=0$

$\mathrm{fps2}≔\mathrm{FPseries}\left(\mathrm{ode2}\,y\left(x\right)\,s\left(n\right)\right)\:$

$\mathrm{sys2}≔\left\{73s\left(0\right)+720s\left(6\right)=80\,s\left(0\right)+s\left(1\right)+s\left(2\right)=3\,s\left(4\right)+s\left(6\right)=\frac{7}{720}\,2s\left(1\right)-s\left(4\right)=2\,s\left(1\right)-s\left(2\right)=0\right\}$

$\mathrm{sys2}≔\left\{73s\left(0\right)+720s\left(6\right)=80\,s\left(1\right)-s\left(2\right)=0\,2s\left(1\right)-s\left(4\right)=2\,s\left(4\right)+s\left(6\right)=\frac{7}{720}\,s\left(0\right)+s\left(1\right)+s\left(2\right)=3\right\}$

$\mathrm{series\_by\_leastsquare}\left(\mathrm{fps2}\,\mathrm{sys2}\right)$

$\mathrm{FPSstruct}\left(1+x+x^2-\frac{x^3}{6}-\frac{x^5}{60}+\frac{7x^6}{720}+\left(\sum _{n=7}^{\mathrm{\infty }}s\left(n\right)x^n\right)\,\left(n^3-3n^2+2n\right)s\left(n\right)+\left(n^3-6n^2+11n-6\right)s\left(n-1\right)+s\left(n-3\right)\right)$