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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

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

Formal power series solution with rational coefficients

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

Formal power series solution with hypergeometric coefficients

 > ode := 2*x*(x-1)*diff(diff(y(x),x),x)+(7*x-3)*diff(y(x),x)+2*y(x) = 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)
 > dsolve(ode,y(x),'type=formal_series','coeffs'='hypergeom');
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{\left({\mathrm{_n}}{+}{1}\right){}{{x}}^{{\mathrm{_n}}}}{{2}{}{\mathrm{_n}}{+}{1}}\right){,}{y}{}\left({x}\right){=}{\mathrm{_C1}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\mathrm{\Gamma }}{}\left(\frac{{1}}{{2}}{+}{\mathrm{_n}}\right){}{\left({x}{+}{1}\right)}^{{\mathrm{_n}}}}{\sqrt{{\mathrm{\pi }}}{}{\mathrm{_n}}{!}}\right){,}{y}{}\left({x}\right){=}\frac{{\mathrm{_C1}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\mathrm{\Gamma }}{}\left(\frac{{1}}{{2}}{+}{\mathrm{_n}}\right){}{\left({-1}\right)}^{{\mathrm{_n}}}{}{\left({x}{-}{1}\right)}^{{\mathrm{_n}}}}{{\mathrm{\Gamma }}{}\left({\mathrm{_n}}{+}{1}\right)}\right)}{\sqrt{{\mathrm{\pi }}}}$ (6)

Formal m-sparse m-hypergeometric power series solutions

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