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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

 > $\mathrm{ode}≔\left(3{x}^{2}-6x+3\right)\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right),x\right),x\right)+\left(12x-12\right)\mathrm{diff}\left(y\left(x\right),x\right)+6y\left(x\right)$
 ${\mathrm{ode}}{≔}\left({3}{}{{x}}^{{2}}{-}{6}{}{x}{+}{3}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({12}{}{x}{-}{12}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{6}{}{y}{}\left({x}\right)$ (1)
 > $\mathrm{dsolve}\left(\mathrm{ode},y\left(x\right),'\mathrm{formal_series}','\mathrm{coeffs}'='\mathrm{polynomial}'\right)$
 ${y}{}\left({x}\right){=}{\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}\left({\mathrm{_C2}}{}{\mathrm{_n}}{+}{\mathrm{_C1}}\right){}{{x}}^{{\mathrm{_n}}}$ (2)

Formal power series solution with rational coefficients

 > $\mathrm{ode}≔\left(3-x\right)\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right),x\right),x\right)-\mathrm{diff}\left(y\left(x\right),x\right)$
 ${\mathrm{ode}}{≔}\left({3}{-}{x}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)$ (3)
 > $\mathrm{dsolve}\left(\mathrm{ode},y\left(x\right),'\mathrm{formal_series}','\mathrm{coeffs}'='\mathrm{rational}'\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C2}}{+}{\mathrm{_C1}}{}\left({\sum }_{{\mathrm{_n}}{=}{1}}^{{\mathrm{\infty }}}{}\frac{{\left({x}{-}{2}\right)}^{{\mathrm{_n}}}}{{\mathrm{_n}}}\right)$ (4)

Formal power series solution with hypergeometric coefficients

 > $\mathrm{ode}≔2x\left(x-1\right)\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right),x\right),x\right)+\left(7x-3\right)\mathrm{diff}\left(y\left(x\right),x\right)+2y\left(x\right)=0$
 ${\mathrm{ode}}{≔}{2}{}{x}{}\left({x}{-}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({7}{}{x}{-}{3}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{2}{}{y}{}\left({x}\right){=}{0}$ (5)
 > $\mathrm{dsolve}\left(\mathrm{ode},y\left(x\right),'\mathrm{type}=\mathrm{formal_series}','\mathrm{coeffs}'='\mathrm{hypergeom}'\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{\left({\mathrm{_n}}{+}{1}\right){}{{x}}^{{\mathrm{_n}}}}{{2}{}{\mathrm{_n}}{+}{1}}\right){,}{y}{}\left({x}\right){=}{\mathrm{_C1}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\mathrm{\Gamma }}{}\left(\frac{{1}}{{2}}{+}{\mathrm{_n}}\right){}{\left({x}{+}{1}\right)}^{{\mathrm{_n}}}}{\sqrt{{\mathrm{\pi }}}{}{\mathrm{_n}}{!}}\right){,}{y}{}\left({x}\right){=}\frac{{\mathrm{_C1}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\mathrm{\Gamma }}{}\left(\frac{{1}}{{2}}{+}{\mathrm{_n}}\right){}{\left({-1}\right)}^{{\mathrm{_n}}}{}{\left({x}{-}{1}\right)}^{{\mathrm{_n}}}}{{\mathrm{\Gamma }}{}\left({\mathrm{_n}}{+}{1}\right)}\right)}{\sqrt{{\mathrm{\pi }}}}$ (6)

Formal m-sparse m-hypergeometric power series solutions

 > $\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right),x,x\right)+\left(x-1\right)y\left(x\right)$
 ${\mathrm{ode}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}\left({x}{-}{1}\right){}{y}{}\left({x}\right)$ (7)
 > $\mathrm{dsolve}\left(\mathrm{ode},y\left(x\right),'\mathrm{type}=\mathrm{formal_series}','\mathrm{coeffs}'='\mathrm{mhypergeom}'\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right){}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({-}\frac{{1}}{{9}}\right)}^{{\mathrm{_n}}}{}{\left({x}{-}{1}\right)}^{{3}{}{\mathrm{_n}}}}{{\mathrm{\Gamma }}{}\left({\mathrm{_n}}{+}{1}\right){}{\mathrm{\Gamma }}{}\left({\mathrm{_n}}{+}\frac{{2}}{{3}}\right)}\right){,}{y}{}\left({x}\right){=}\frac{{2}{}{\mathrm{_C1}}{}{\mathrm{\pi }}{}\sqrt{{3}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({-}\frac{{1}}{{9}}\right)}^{{\mathrm{_n}}}{}{\left({x}{-}{1}\right)}^{{3}{}{\mathrm{_n}}{+}{1}}}{{\mathrm{\Gamma }}{}\left({\mathrm{_n}}{+}\frac{{4}}{{3}}\right){}{\mathrm{\Gamma }}{}\left({\mathrm{_n}}{+}{1}\right)}\right)}{{9}{}{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right)}$ (8)