When the input ODE is a linear ode with polynomial coefficients which is homogeneous or inhomogeneous with rational right hand side, and the optional arguments 'formal_series' (or 'type=formal_series') and 'coeffs'=coeff_type are given, dsolve will return a set of formal power series solutions with the specified coefficients at all candidate points of expansion. See Slode for more details.

$\mathrm{ode}≔\left(3x^2-6x+3\right)\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)\,x\right)+\left(12x-12\right)\mathrm{diff}\left(y\left(x\right)\,x\right)+6y\left(x\right)$

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

$\mathrm{dsolve}\left(\mathrm{ode}\,y\left(x\right)\,'\mathrm{formal\_series}'\,'\mathrm{coeffs}'='\mathrm{polynomial}'\right)$

$y\left(x\right)=\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\left(\mathrm{\_C2}\mathrm{\_n}+\mathrm{\_C1}\right)x^{\mathrm{\_n}}$

$\mathrm{ode}≔\left(3-x\right)\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)\,x\right)-\mathrm{diff}\left(y\left(x\right)\,x\right)$

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

$\mathrm{dsolve}\left(\mathrm{ode}\,y\left(x\right)\,'\mathrm{formal\_series}'\,'\mathrm{coeffs}'='\mathrm{rational}'\right)$

$y\left(x\right)=\mathrm{\_C2}+\mathrm{\_C1}\left(\sum _{\mathrm{\_n}=1}^{\mathrm{\infty }}\frac{{\left(x-2\right)}^{\mathrm{\_n}}}{\mathrm{\_n}}\right)$

$\mathrm{ode}≔2x\left(x-1\right)\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)\,x\right)+\left(7x-3\right)\mathrm{diff}\left(y\left(x\right)\,x\right)+2y\left(x\right)=0$

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

$\mathrm{dsolve}\left(\mathrm{ode}\,y\left(x\right)\,'\mathrm{type}=\mathrm{formal\_series}'\,'\mathrm{coeffs}'='\mathrm{hypergeom}'\right)$

$y\left(x\right)=\mathrm{\_C1}\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\frac{\left(\mathrm{\_n}+1\right)x^{\mathrm{\_n}}}{2\mathrm{\_n}+1}\right),y\left(x\right)=\frac{\mathrm{\_C1}\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\frac{1}{2}+\mathrm{\_n}\right){\left(x+1\right)}^{\mathrm{\_n}}}{\mathrm{\_n}!}\right)}{\sqrt{\mathrm{\pi }}},y\left(x\right)=\frac{\mathrm{\_C1}\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\frac{1}{2}+\mathrm{\_n}\right){\left(\mathrm{-1}\right)}^{\mathrm{\_n}}{\left(x-1\right)}^{\mathrm{\_n}}}{\mathrm{\Gamma }\left(\mathrm{\_n}+1\right)}\right)}{\sqrt{\mathrm{\pi }}}$

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

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

$\mathrm{dsolve}\left(\mathrm{ode}\,y\left(x\right)\,'\mathrm{type}=\mathrm{formal\_series}'\,'\mathrm{coeffs}'='\mathrm{mhypergeom}'\right)$

$y\left(x\right)=\mathrm{\_C1}\mathrm{\Gamma }\left(\frac{2}{3}\right)\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{9}\right)}^{\mathrm{\_n}}{\left(x-1\right)}^{3\mathrm{\_n}}}{\mathrm{\Gamma }\left(\mathrm{\_n}+1\right)\mathrm{\Gamma }\left(\mathrm{\_n}+\frac{2}{3}\right)}\right),y\left(x\right)=\frac{2\mathrm{\_C1}\mathrm{\pi }\sqrt{3}\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{9}\right)}^{\mathrm{\_n}}{\left(x-1\right)}^{3\mathrm{\_n}+1}}{\mathrm{\Gamma }\left(\mathrm{\_n}+\frac{4}{3}\right)\mathrm{\Gamma }\left(\mathrm{\_n}+1\right)}\right)}{9\mathrm{\Gamma }\left(\frac{2}{3}\right)}$