Jacobi - Maple Help

Jacobi ODEs

Description

 • The general form of the Jacobi ODE is given by the following:
 > Jacobi_ode := diff(y(x),x,x)*x*(1-x) = (g-(a+1)*x)*diff(y(x),x)+n*(a+n)*y(x);
 ${\mathrm{Jacobi_ode}}{≔}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{x}{}\left({1}{-}{x}\right){=}\left({g}{-}\left({a}{+}{1}\right){}{x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{n}{}\left({a}{+}{n}\right){}{y}{}\left({x}\right)$ (1)
 where n is an integer. See Iyanaga and Kawada, "Encyclopedic Dictionary of Mathematics", p. 1480.

Examples

The solution to this type of ODE can be expressed in terms of the hypergeometric function; see hypergeom.

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$
 $\left[{\mathrm{odeadvisor}}\right]$ (2)
 > $\mathrm{odeadvisor}\left(\mathrm{Jacobi_ode}\right)$
 $\left[{\mathrm{_Jacobi}}\right]$ (3)
 > $\mathrm{dsolve}\left(\mathrm{Jacobi_ode}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{hypergeom}}{}\left(\left[{-}{1}{-}\frac{{a}}{{2}}{-}\frac{\sqrt{{{a}}^{{2}}{+}\left({-}{4}{}{n}{+}{4}\right){}{a}{-}{4}{}{{n}}^{{2}}{+}{4}}}{{2}}{,}{-}{1}{-}\frac{{a}}{{2}}{+}\frac{\sqrt{{{a}}^{{2}}{+}\left({-}{4}{}{n}{+}{4}\right){}{a}{-}{4}{}{{n}}^{{2}}{+}{4}}}{{2}}\right]{,}\left[{-}{g}\right]{,}{x}\right){+}{\mathrm{_C2}}{}{{x}}^{{1}{+}{g}}{}{\mathrm{hypergeom}}{}\left(\left[{-}\frac{{a}}{{2}}{-}\frac{\sqrt{{{a}}^{{2}}{+}\left({-}{4}{}{n}{+}{4}\right){}{a}{-}{4}{}{{n}}^{{2}}{+}{4}}}{{2}}{+}{g}{,}{-}\frac{{a}}{{2}}{+}\frac{\sqrt{{{a}}^{{2}}{+}\left({-}{4}{}{n}{+}{4}\right){}{a}{-}{4}{}{{n}}^{{2}}{+}{4}}}{{2}}{+}{g}\right]{,}\left[{2}{+}{g}\right]{,}{x}\right)$ (4)
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 7, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9878969192504883, "perplexity": 1698.6907039627558}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358233.7/warc/CC-MAIN-20211127193525-20211127223525-00560.warc.gz"}