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dsolve

Find formal power series solutions to a linear ODE with polynomial coefficients

 Calling Sequence dsolve(ODE, y(x), 'formal_series', 'coeffs'=coeff_type) dsolve(ODE, y(x), 'type=formal_series', 'coeffs'=coeff_type)

Parameters

 ODE - linear ordinary differential equation with polynomial coefficients y(x) - the dependent variable (the indeterminate function) 'type=formal_series' - request for formal power series solutions 'coeffs'=coeff_type - coeff_type is one of 'polynomial', 'rational', 'hypergeom', 'mhypergeom'

Description

 • 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.

Examples

Formal power series solution with polynomial coefficients

 > $\mathrm{ode}≔\left(3{x}^{2}-6x+3\right)\left(\frac{ⅆ}{ⅆx}\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)\right)+\left(12x-12\right)\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+6y\left(x\right)$
 ${\mathrm{ode}}{:=}\left({3}{}{{x}}^{{2}}{-}{6}{}{x}{+}{3}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right)\right){+}\left({12}{}{x}{-}{12}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){+}{6}{}{y}{}\left({x}\right)$ (1)
 > $\mathrm{dsolve}\left(\mathrm{ode},y\left(x\right),'\mathrm{formal_series}','\mathrm{coeffs}'='\mathrm{polynomial}'\right)$
 ${y}{}\left({x}\right){=}{\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\mathrm{_C2}}{}{\mathrm{_n}}{+}{\mathrm{_C1}}\right){}{{x}}^{{\mathrm{_n}}}$ (2)

Formal power series solution with rational coefficients

 > $\mathrm{ode}≔\left(3-x\right)\left(\frac{ⅆ}{ⅆx}\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)\right)-\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)$
 ${\mathrm{ode}}{:=}\left({3}{-}{x}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right)\right){-}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right)$ (3)
 > $\mathrm{dsolve}\left(\mathrm{ode},y\left(x\right),'\mathrm{formal_series}','\mathrm{coeffs}'='\mathrm{rational}'\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C2}}{+}{\mathrm{_C1}}{}\left({\sum }_{{\mathrm{_n}}{=}{1}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left({x}{-}{2}\right)}^{{\mathrm{_n}}}}{{\mathrm{_n}}}\right)$ (4)

Formal power series solution with hypergeometric coefficients

 > $\mathrm{ode}≔2x\left(x-1\right)\left(\frac{ⅆ}{ⅆx}\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)\right)+\left(7x-3\right)\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+2y\left(x\right)=0$
 ${\mathrm{ode}}{:=}{2}{}{x}{}\left({x}{-}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right)\right){+}\left({7}{}{x}{-}{3}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){+}{2}{}{y}{}\left({x}\right){=}{0}$ (5)
 > $\mathrm{dsolve}\left(\mathrm{ode},y\left(x\right),'\mathrm{type}=\mathrm{formal_series}','\mathrm{coeffs}'='\mathrm{hypergeom}'\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{\left({\mathrm{_n}}{+}{1}\right){}{{x}}^{{\mathrm{_n}}}}{{2}{}{\mathrm{_n}}{+}{1}}\right){,}{y}{}\left({x}\right){=}{\mathrm{_C1}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{+}{\mathrm{_n}}\right){}{\left({x}{+}{1}\right)}^{{\mathrm{_n}}}}{\sqrt{{\mathrm{π}}}{}{\mathrm{_n}}{!}}\right){,}{y}{}\left({x}\right){=}\frac{{\mathrm{_C1}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\mathrm{Γ}}{}\left(\frac{{1}}{{2}}{+}{\mathrm{_n}}\right){}{\left({-}{1}\right)}^{{\mathrm{_n}}}{}{\left({x}{-}{1}\right)}^{{\mathrm{_n}}}}{{\mathrm{Γ}}{}\left({\mathrm{_n}}{+}{1}\right)}\right)}{\sqrt{{\mathrm{π}}}}$ (6)

Formal m-sparse m-hypergeometric power series solutions

 > $\mathrm{ode}≔\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)+\left(x-1\right)y\left(x\right)$
 ${\mathrm{ode}}{:=}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right){+}\left({x}{-}{1}\right){}{y}{}\left({x}\right)$ (7)
 > $\mathrm{dsolve}\left(\mathrm{ode},y\left(x\right),'\mathrm{type}=\mathrm{formal_series}','\mathrm{coeffs}'='\mathrm{mhypergeom}'\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right){}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left({-}\frac{{1}}{{9}}\right)}^{{\mathrm{_n}}}{}{\left({x}{-}{1}\right)}^{{3}{}{\mathrm{_n}}}}{{\mathrm{Γ}}{}\left({\mathrm{_n}}{+}{1}\right){}{\mathrm{Γ}}{}\left({\mathrm{_n}}{+}\frac{{2}}{{3}}\right)}\right){,}{y}{}\left({x}\right){=}\frac{{2}}{{9}}{}\frac{{\mathrm{_C1}}{}{\mathrm{π}}{}\sqrt{{3}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left({-}\frac{{1}}{{9}}\right)}^{{\mathrm{_n}}}{}{\left({x}{-}{1}\right)}^{{3}{}{\mathrm{_n}}{+}{1}}}{{\mathrm{Γ}}{}\left({\mathrm{_n}}{+}\frac{{4}}{{3}}\right){}{\mathrm{Γ}}{}\left({\mathrm{_n}}{+}{1}\right)}\right)}{{\mathrm{Γ}}{}\left(\frac{{2}}{{3}}\right)}$ (8)