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dsolve/formal_solution

find formal solutions to a homogeneous linear ODE with polynomial coefficients

 Calling Sequence dsolve(ODE, y(x), 'formal_solution', 'coeffs'=coeff_type, 'point'=x0) dsolve(ODE, y(x), 'type=formal_solution', 'coeffs'=coeff_type, 'point'=x0)

Parameters

 ODE - homogeneous linear ordinary differential equation with polynomial coefficients y(x) - dependent variable (the indeterminate function) 'type=formal_solution' - (optional) request for formal solutions 'coeffs'=coeff_type - (optional) coeff_type is one of 'mhypergeom', 'dAlembertian' 'point'=x0 - algebraic number, rational in parameters, or infinity

Description

 • When the input ODE is a homogeneous linear ode with polynomial coefficients, and the optional arguments 'formal_solution' (or 'type=formal_solution') and 'coeffs'=coeff_type are given, the dsolve command returns a set of formal solutions with the specified coefficients at the given point (the default is at the origin). For more information, see Slode[mhypergeom_formal_sol] and Slode[dAlembertian_formal_sol].

Examples

Find the formal solution set with m-hypergeometric series coefficients.

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

Find the formal solution set with d'Alembertian series coefficient.

 > $\mathrm{ode}≔\left(-4-{x}^{2}+2x\right)y\left(x\right)+\left(2x-3{x}^{3}-{x}^{2}\right)\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+\left({x}^{3}-{x}^{4}\right)\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)$
 ${\mathrm{ode}}{≔}\left({-}{{x}}^{{2}}{+}{2}{}{x}{-}{4}\right){}{y}{}\left({x}\right){+}\left({-}{3}{}{{x}}^{{3}}{-}{{x}}^{{2}}{+}{2}{}{x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({-}{{x}}^{{4}}{+}{{x}}^{{3}}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)$ (3)
 > $\mathrm{dsolve}\left(\mathrm{ode},y\left(x\right),'\mathrm{formal_solution}','\mathrm{coeffs}'='\mathrm{dAlembertian}'\right)$
 ${y}{}\left({x}\right){=}{{x}}^{{2}}{}\left({-}\frac{\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}{{x}}^{{\mathrm{_n}}}\right)}{{2}}{+}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}\left({\sum }_{{\mathrm{_n1}}{=}{0}}^{{\mathrm{_n}}{-}{1}}{}{\left({-}\frac{{1}}{{2}}\right)}^{{\mathrm{_n1}}}{}\left({\prod }_{{\mathrm{_k}}{=}{0}}^{{\mathrm{_n1}}{-}{1}}{}\frac{{\left({\mathrm{_k}}{+}{3}\right)}^{{2}}}{{\mathrm{_k}}{+}{2}}\right)\right){}{{x}}^{{\mathrm{_n}}}\right)\right){}{\mathrm{_C1}}{+}\frac{{{ⅇ}}^{\frac{{2}}{{x}}}{}\left(\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}{{x}}^{{\mathrm{_n}}}\right){-}\frac{{1}}{{3}}\right){}{\mathrm{_C2}}}{{x}}$ (4)
 > $\mathrm{ode}≔-\left(x-1\right)y\left(x\right)-\left(2{x}^{2}-4x-1\right)\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)-\frac{1x\left(x+1\right)\left(x-6\right)\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)}{2}+\frac{1\left(2+x\right){x}^{2}\left(\frac{{ⅆ}^{3}}{ⅆ{x}^{3}}y\left(x\right)\right)}{2}$
 ${\mathrm{ode}}{≔}{-}\left({x}{-}{1}\right){}{y}{}\left({x}\right){-}\left({2}{}{{x}}^{{2}}{-}{4}{}{x}{-}{1}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}\frac{{x}{}\left({x}{+}{1}\right){}\left({x}{-}{6}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{{2}}{+}\frac{\left({2}{+}{x}\right){}{{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{{2}}$ (5)
 > $\mathrm{dsolve}\left(\mathrm{ode},y\left(x\right),'\mathrm{formal_solution}','\mathrm{coeffs}'='\mathrm{dAlembertian}','\mathrm{point}'=a\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}{\left({-}\frac{{1}}{{a}{+}{2}}\right)}^{{\mathrm{_n}}}{}\left({\prod }_{{\mathrm{_k}}{=}{0}}^{{\mathrm{_n}}{-}{1}}{}\frac{{\mathrm{_k}}{+}{2}}{{\mathrm{_k}}{+}{1}}\right){}{\left({x}{-}{a}\right)}^{{\mathrm{_n}}}\right){+}{\mathrm{_C2}}{}\left({-}{a}{-}{2}\right){}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}{\left({-}\frac{{1}}{{a}{+}{2}}\right)}^{{\mathrm{_n}}}{}\left({\prod }_{{\mathrm{_k}}{=}{0}}^{{\mathrm{_n}}{-}{1}}{}\frac{{\mathrm{_k}}{+}{2}}{{\mathrm{_k}}{+}{1}}\right){}\left({\sum }_{{\mathrm{_n1}}{=}{0}}^{{\mathrm{_n}}{-}{1}}{}\frac{\left({\mathrm{_n1}}{+}{1}\right){}{\left(\frac{{a}{+}{2}}{{a}}\right)}^{{\mathrm{_n1}}}{}\left({\prod }_{{\mathrm{_k}}{=}{0}}^{{\mathrm{_n1}}{-}{1}}{}\frac{{\left({\mathrm{_k}}{+}{1}\right)}^{{2}}}{{\left({\mathrm{_k}}{+}{2}\right)}^{{2}}}\right)}{{\mathrm{_n1}}{+}{2}}\right){}{\left({x}{-}{a}\right)}^{{\mathrm{_n}}}\right){+}\left({3}{}{a}{}\left({a}{+}{2}\right){}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}{\left({-}\frac{{1}}{{a}{+}{2}}\right)}^{{\mathrm{_n}}}{}\left({\prod }_{{\mathrm{_k}}{=}{0}}^{{\mathrm{_n}}{-}{1}}{}\frac{{\mathrm{_k}}{+}{2}}{{\mathrm{_k}}{+}{1}}\right){}\left({\sum }_{{\mathrm{_n1}}{=}{0}}^{{\mathrm{_n}}{-}{1}}{}\frac{\left({\mathrm{_n1}}{+}{1}\right){}{\left(\frac{{a}{+}{2}}{{a}}\right)}^{{\mathrm{_n1}}}{}\left({\prod }_{{\mathrm{_k}}{=}{0}}^{{\mathrm{_n1}}{-}{1}}{}\frac{{\left({\mathrm{_k}}{+}{1}\right)}^{{2}}}{{\left({\mathrm{_k}}{+}{2}\right)}^{{2}}}\right){}\left({\sum }_{{\mathrm{_n2}}{=}{0}}^{{\mathrm{_n1}}{-}{1}}{}\frac{\left({\mathrm{_n2}}{+}{2}\right){}\left({\prod }_{{\mathrm{_k}}{=}{0}}^{{\mathrm{_n2}}{-}{1}}{}\frac{{\mathrm{_k}}{+}{2}}{{\mathrm{_k}}{+}{3}}\right)}{{\mathrm{_n2}}{+}{1}}\right)}{{\mathrm{_n1}}{+}{2}}\right){}{\left({x}{-}{a}\right)}^{{\mathrm{_n}}}\right){-}{{a}}^{{2}}{}\left({a}{+}{2}\right){}\left({\sum }_{{\mathrm{_n}}{=}{0}}^{{\mathrm{\infty }}}{}{\left({-}\frac{{1}}{{a}{+}{2}}\right)}^{{\mathrm{_n}}}{}\left({\prod }_{{\mathrm{_k}}{=}{0}}^{{\mathrm{_n}}{-}{1}}{}\frac{{\mathrm{_k}}{+}{2}}{{\mathrm{_k}}{+}{1}}\right){}\left({\sum }_{{\mathrm{_n1}}{=}{0}}^{{\mathrm{_n}}{-}{1}}{}\frac{\left({\mathrm{_n1}}{+}{1}\right){}{\left(\frac{{a}{+}{2}}{{a}}\right)}^{{\mathrm{_n1}}}{}\left({\prod }_{{\mathrm{_k}}{=}{0}}^{{\mathrm{_n1}}{-}{1}}{}\frac{{\left({\mathrm{_k}}{+}{1}\right)}^{{2}}}{{\left({\mathrm{_k}}{+}{2}\right)}^{{2}}}\right){}\left({\sum }_{{\mathrm{_n2}}{=}{0}}^{{\mathrm{_n1}}{-}{1}}{}\frac{\left({\mathrm{_n2}}{+}{2}\right){}\left({\prod }_{{\mathrm{_k}}{=}{0}}^{{\mathrm{_n2}}{-}{1}}{}\frac{{\mathrm{_k}}{+}{2}}{{\mathrm{_k}}{+}{3}}\right){}\left({\sum }_{{\mathrm{_n3}}{=}{0}}^{{\mathrm{_n2}}{-}{1}}{}\frac{\left({\mathrm{_n3}}{+}{3}\right){}{\left({-}{a}\right)}^{{\mathrm{_n3}}}{}\left({\prod }_{{\mathrm{_k}}{=}{0}}^{{\mathrm{_n3}}{-}{1}}{}\frac{{\mathrm{_k}}{+}{3}}{\left({\mathrm{_k}}{+}{4}\right){}\left({\mathrm{_k}}{+}{1}\right)}\right)}{{\mathrm{_n3}}{+}{1}}\right)}{{\mathrm{_n2}}{+}{1}}\right)}{{\mathrm{_n1}}{+}{2}}\right){}{\left({x}{-}{a}\right)}^{{\mathrm{_n}}}\right)\right){}{\mathrm{_C3}}$ (6)