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$\mathrm{ode}:=\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\left(x-1\right)-y\left(x\right)\=0$

$\mathrm{ode}:=\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\left(x-1\right)-y\left(x\right)\=0$

$\mathrm{DEdetermine}\left(\mathrm{ode}\,y\left(x\right)\right)$

$\mathrm{LODEstruct}\left(\left\{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\left(x-1\right)-y\left(x\right)\=0\right\}\,\left\{y\left(x\right)\right\}\right)$

$\mathrm{ode}:=\left(3x^2-6x\+3\right)\left(\frac{\ⅆ}{\ⅆx}\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\right)\+\left(12x-12\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\+6y\left(x\right)$

$\mathrm{ode}:=\left(3x^2-6x\+3\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)\+\left(12x-12\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\+6y\left(x\right)$

$\mathrm{FPseries}\left(\mathrm{ode}\,y\left(x\right)\,v\left(n\right)\right)$

$\mathrm{FPSstruct}\left({\mathrm{\_C}}_0\+{\mathrm{\_C}}_{1}x\+\sum _{n\=2}^{\mathrm{\∞}}v\left(n\right)x^n\,\left(n^2-n\right)v\left(n\right)\+\left(-2n^2\+2n\right)v\left(n-1\right)\+\left(n^2-n\right)v\left(n-2\right)\right)$

$\mathrm{FTseries}\left(\mathrm{ode}\,y\left(x\right)\,v\left(n\right)\,0\,\mathrm{\_A}\,2\right)$

$\mathrm{FPSstruct}\left({\mathrm{\_A}}_0\+{\mathrm{\_A}}_{1}x\+\frac{1}{2}\left(-2{\mathrm{\_A}}_0\+4{\mathrm{\_A}}_1\right)x^2\+\sum _{n\=3}^{\mathrm{\∞}}\frac{v\left(n\right)x^n}{n\!}\,v\left(n\right)-2nv\left(n-1\right)\+\left(n^2-n\right)v\left(n-2\right)\right)$

$\mathrm{ode}:=\left(3x^2-6x\+3\right)\left(\frac{\ⅆ}{\ⅆx}\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\right)\+\left(12x-12\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\+6y\left(x\right)$

$\mathrm{ode}:=\left(3x^2-6x\+3\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)\+\left(12x-12\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\+6y\left(x\right)$

$\mathrm{candidate\_points}\left(\mathrm{ode}\,y\left(x\right)\,\'\mathrm{polynomial}\'\right)$

$\left\{0\right\}$

$\mathrm{candidate\_points}\left(\mathrm{ode}\,y\left(x\right)\,\'\mathrm{rational}\'\right)$

$\left\{1\right\}$

$\mathrm{candidate\_points}\left(\mathrm{ode}\,y\left(x\right)\,\'\mathrm{hypergeom}\'\right)$

$\left\{1\,\mathrm{any\_ordinary\_point}\right\}$

$\mathrm{ode}:=\left(2\+x^2\right)\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)-2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)x\+\left(2\+x^2\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-2xy\left(x\right)$

$\mathrm{ode}:=\left(2\+x^2\right)\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)-2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)x\+\left(2\+x^2\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-2xy\left(x\right)$

$\mathrm{candidate\_mpoints}\left(\mathrm{ode}\,y\left(x\right)\right)$

$\left[\left[2\,\mathrm{LODEstruct}\left(\left\{y\left(x\right)\+\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right\}\,\left\{y\left(x\right)\right\}\right)\,\left\{0\right\}\right]\right]$

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

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

$\mathrm{candidate\_mpoints}\left(\mathrm{ode}\,y\left(x\right)\,\'\mathrm{irreducible}\'\right)$

$\left[\left[3\,\mathrm{LODEstruct}\left(\left\{y\left(x\right)\right\}\,\left\{y\left(x\right)\right\}\right)\,\left\{1\right\}\right]\right]$

$\mathrm{ode}:=\left(3x^2-6x\+3\right)\left(\frac{\ⅆ}{\ⅆx}\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\right)\+\left(12x-12\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\+6y\left(x\right)$

$\mathrm{ode}:=\left(3x^2-6x\+3\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)\+\left(12x-12\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\+6y\left(x\right)$

$\mathrm{polynomial\_series\_sol}\left(\mathrm{ode}\,y\left(x\right)\right)$

$\left\{\sum _{n\=0}^{\mathrm{\∞}}\left({\mathrm{\_C}}_0\+n{\mathrm{\_C}}_1\right)x^n\right\}$

$\mathrm{ode}:=\left(3-x\right)\left(\frac{\ⅆ}{\ⅆx}\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\right)-\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)$

$\mathrm{ode}:=\left(3-x\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)$

$\mathrm{rational\_series\_sol}\left(\mathrm{ode}\,y\left(x\right)\right)$

$\left\{{\mathrm{\_C}}_1\+{\mathrm{\_C}}_0\left(\sum _{n\=1}^{\mathrm{\∞}}\frac{{\left(x-2\right)}^n}{n}\right)\right\}$

$\mathrm{ode}:=2x\left(x-1\right)\left(\frac{\ⅆ}{\ⅆx}\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\right)\+\left(7x-3\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\+2y\left(x\right)\=0$

$\mathrm{ode}:=2x\left(x-1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)\+\left(7x-3\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\+2y\left(x\right)\=0$

$\mathrm{hypergeom\_series\_sol}\left(\mathrm{ode}\,y\left(x\right)\right)$

$\left\{{\mathrm{\_C}}_1\left(\sum _{n\=0}^{\mathrm{\∞}}\frac{\left(n\+1\right)x^n}{2n\+1}\right)\,{\mathrm{\_C}}_1\left(\sum _{n\=0}^{\mathrm{\∞}}\frac{{\left(-1\right)}^n\mathrm{\Γ}\left(n\+\frac{1}{2}\right){\left(x-1\right)}^n}{\mathrm{\Γ}\left(n\+1\right)}\right)\,{\mathrm{\_C}}_1\left(\sum _{n\=0}^{\mathrm{\∞}}\frac{\mathrm{\Γ}\left(n\+\frac{1}{2}\right){\left(x\+1\right)}^n}{n\!}\right)\right\}$

$\mathrm{ode}:=\frac{2\left(x^3-\frac{2x^2}{3}\+\frac{1x}{9}\right)\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)}{9}\+\frac{2\left(9x^2-4x\+\frac{1}{3}\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)}{9}\+\frac{2\left(18x-4\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}{9}\+\frac{4y\left(x\right)}{3}$

$\mathrm{ode}:=\frac{1}{9}\left(2x^3-\frac{4}{3}{x}^2\+\frac{2}{9}x\right)\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)\+\frac{1}{9}\left(18x^2-8x\+\frac{2}{3}\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)\+\frac{1}{9}\left(36x-8\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\+\frac{4}{3}y\left(x\right)$

$\mathrm{msparse\_series\_sol}\left(\mathrm{ode}\,y\left(x\right)\,v\left(n\right)\right)$

$\left\{\mathrm{FPSstruct}\left(\frac{1}{36}{\mathrm{\_C}}_2\+{\mathrm{\_C}}_2{\left(x-\frac{1}{6}\right)}^2\+36{\mathrm{\_C}}_2{\left(x-\frac{1}{6}\right)}^4\+\sum _{n\=3}^{\mathrm{\∞}}v\left(2n\right){\left(x-\frac{1}{6}\right)}^{2n}\,-36v\left(2n-2\right)\+v\left(2n\right)\right)\,\mathrm{FPSstruct}\left({\mathrm{\_C}}_1\left(x-\frac{1}{6}\right)\+36{\mathrm{\_C}}_1{\left(x-\frac{1}{6}\right)}^3\+1296{\mathrm{\_C}}_1{\left(x-\frac{1}{6}\right)}^5\+\sum _{n\=3}^{\mathrm{\∞}}v\left(2n\+1\right){\left(x-\frac{1}{6}\right)}^{2n\+1}\,-36v\left(2n-1\right)\+v\left(2n\+1\right)\right)\right\}$

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

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

$\mathrm{mhypergeom\_series\_sol}\left(\mathrm{ode}\,y\left(x\right)\right)$

$\left\{{\mathrm{\_C}}_1\left(\sum _{n\=0}^{\mathrm{\∞}}\frac{{\left(-\frac{1}{9}\right)}^n{\left(x-1\right)}^{3n\+1}}{\mathrm{\Γ}\left(n\+1\right)\mathrm{\Γ}\left(n\+\frac{4}{3}\right)}\right)\,{\mathrm{\_C}}_1\left(\sum _{n\=0}^{\mathrm{\∞}}\frac{{\left(-\frac{1}{9}\right)}^n{\left(x-1\right)}^{3n}}{\mathrm{\Γ}\left(n\+\frac{2}{3}\right)\mathrm{\Γ}\left(n\+1\right)}\right)\right\}$

$\mathrm{ode}:=3xy\left(x\right)\+\left(4\+4x\right)x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\+\left(-3-3x\right)x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)$

$\mathrm{ode}:=3xy\left(x\right)\+\left(4\+4x\right)x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\+\left(-3-3x\right)x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)$

$\mathrm{hypergeom\_formal\_sol}\left(\mathrm{ode}\,y\left(x\right)\,t\,0\right)$

$\left[\left[x\left(t\right)\=t\,y\left(t\right)\=\frac{9}{4}\frac{\mathrm{\Γ}\left(\frac{2}{3}\right)\left(\sum _{n\=0}^{\mathrm{\∞}}\frac{{\left(-1\right)}^{n\+1}\mathrm{\Γ}\left(n-\frac{7}{6}\+\frac{1}{6}\sqrt{85}\right)\mathrm{\Γ}\left(n-\frac{7}{6}-\frac{1}{6}\sqrt{85}\right)t^n}{\mathrm{\Γ}\left(n-\frac{4}{3}\right)\mathrm{\Γ}\left(n\+1\right)}\right)}{\mathrm{\Γ}\left(-\frac{1}{6}\+\frac{1}{6}\sqrt{85}\right)\mathrm{\Γ}\left(-\frac{1}{6}-\frac{1}{6}\sqrt{85}\right)}\right]\,\left[x\left(t\right)\=t\,y\left(t\right)\=\frac{56}{81}\frac{t^{7\/3}\sqrt{3}\mathrm{\π}\left(\sum _{n\=0}^{\mathrm{\∞}}\frac{{\left(-1\right)}^{n\+1}\mathrm{\Γ}\left(n\+\frac{7}{6}\+\frac{1}{6}\sqrt{85}\right)\mathrm{\Γ}\left(n\+\frac{7}{6}-\frac{1}{6}\sqrt{85}\right)t^n}{\mathrm{\Γ}\left(n\+\frac{10}{3}\right)\mathrm{\Γ}\left(n\+1\right)}\right)}{\mathrm{\Γ}\left(\frac{13}{6}\+\frac{1}{6}\sqrt{85}\right)\mathrm{\Γ}\left(\frac{13}{6}-\frac{1}{6}\sqrt{85}\right)\mathrm{\Γ}\left(\frac{2}{3}\right)}\right]\right]$