 ODEs and PDEs - Maple Help

 Ordinary and Partial Differential Equations

Maple is the world leader in finding exact solutions to ordinary and partial differential equations. Maple 2021 extends that lead even further with new algorithms and techniques for solving more ODEs and PDEs.

For Maple 2021, there are significant improvements in dsolve for the exact solution of ${2}^{\mathrm{nd}}$order linear ODEs using hypergeometric functions. The algorithms implemented are at the frontier of the understanding of this problem, and handle classes of extended equations with apparent singularities as well as the most common linear equations with 4 and 5 singularities.

For ODEs and PDEs, the LieAlgebrasOfVectorFields package in Maple 2021 has a new command MapDE , for analyzing the possible linearization of polynomially nonlinear equations, and determining the corresponding mapping when it exists, allowing in that way for the computation of more approximate and exact solutions.

The new Student[ODEs] package covers the material in a standard first course in ODEs and provides step-by-step tools for solving ODEs as well as interactive visualization.  For details, see Student Packages. Advanced type of Hypergeometric solutions for 2nd order linear ODEs

When they exist, 2F1, 1F1 and 0F1 hypergeometric solutions for ${2}^{\mathrm{nd}}$order linear ODEs are now computable in a rather general case. An equation with 3 regular singularities and any number of apparent singularities can now be solved in terms of 2F1 equations using the algorithms presented in (6) and (7). Likewise, an equation with 2 or 1 singularities, one of which is irregular, can now be solved in general using 1F1 and 0F1 hypergeometric functions using the algorithms presented in (1), (2) and (3). Common equations with up to 5 regularities (plus any number of apparent singularities) can now be solved using mappings and the tables from references (4) and (5). All this significantly extends the previous capabilities in solving linear equations, and also of higher order and nonlinear ODE and PDE that through a reduction or symmetry process require the solving of intermediate ${2}^{\mathrm{nd}}$order linear ODEs.

The new algorithms are now automatically used by dsolve and also accessible through a new DEtools command, hypergeometricsols.

Examples

Despite the powerful algorithms for hypergeometric solutions of previous Maple releases, none of the following examples could be solved. Now they can. This equation admits 2F1 solutions where the argument is polynomial in x of degree 3

 >
 > $\mathrm{dsolve}\left(\mathrm{ode__1}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{}_{{2}}{F}_{{1}}{}\left(\frac{{1}}{{12}}{-}\frac{{a}}{{2}}{,}{-}\frac{{a}}{{2}}{+}\frac{{5}}{{12}}{;}\frac{{1}}{{2}}{;}{-}{b}{}\left({x}{-}{1}\right){}{{x}}^{{2}}\right){+}{\mathrm{_C2}}{}{x}{}\sqrt{{1}{-}{x}}{}{}_{{2}}{F}_{{1}}{}\left(\frac{{7}}{{12}}{-}\frac{{a}}{{2}}{,}\frac{{11}}{{12}}{-}\frac{{a}}{{2}}{;}\frac{{3}}{{2}}{;}{-}{b}{}\left({x}{-}{1}\right){}{{x}}^{{2}}\right)$ (1.1)

Verify this solution

 > $\mathrm{odetest}\left(,\mathrm{ode__1}\right)$
 ${0}$ (1.2)

This other equation admits solutions in terms of modified Bessel functions of the first and second kind

 >
 > $\mathrm{dsolve}\left(\mathrm{ode__2}\right)$
 ${y}{}\left({x}\right){=}\frac{{\mathrm{_C1}}{}{x}{}\left(\left({4}{}{{x}}^{{3}}{-}{36}{}{{x}}^{{2}}{+}{141}{}{x}{-}{225}\right){}{{I}}_{{1}}{}\left(\frac{{x}}{{2}}\right){+}\left({-}{4}{}{{x}}^{{3}}{+}{24}{}{{x}}^{{2}}{-}{63}{}{x}{+}{27}\right){}{{I}}_{{2}}{}\left(\frac{{x}}{{2}}\right)\right)}{{x}{-}{3}}{+}\frac{{\mathrm{_C2}}{}{x}{}\left(\left({-}{4}{}{{x}}^{{3}}{+}{36}{}{{x}}^{{2}}{-}{141}{}{x}{+}{225}\right){}{{K}}_{{1}}{}\left(\frac{{x}}{{2}}\right){+}\left({-}{4}{}{{x}}^{{3}}{+}{24}{}{{x}}^{{2}}{-}{63}{}{x}{+}{27}\right){}{{K}}_{{2}}{}\left(\frac{{x}}{{2}}\right)\right)}{{x}{-}{3}}$ (1.3)

Moreover, note that each independent solution involves a linear combination of functions. Indeed, while Bessel (or more generally, 0F1) solutions are related to equations with 1 singularity of irregular kind, this example has other regular singularities:

 > $\mathrm{DEtools}\left[\mathrm{singularities}\right]\left(\mathrm{ode__2}\right)$
 ${\mathrm{regular}}{=}\left\{{0}{,}{3}{,}\frac{{9}}{{2}}{,}\frac{{15}}{{2}}\right\}{,}{\mathrm{irregular}}{=}\left\{{\mathrm{\infty }}\right\}$ (1.4)

thus requiring the use of a linear combination of pFq functions to construct a solution. A similar situation happens with the next example where, due to the presence of apparent singularities, a linear combination of - this time 1F1 - hypergeometric functions (KummerM and KummerU) is required to solve the problem

 >
 > $\mathrm{dsolve}\left(\mathrm{ode__3}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}\left({12}{}{\left({x}{-}\frac{{1}}{{2}}\right)}^{{2}}{}{M}{}\left(\frac{{5}}{{4}}{,}{3}{,}{x}\right){+}\left({-}{20}{}{x}{+}{5}\right){}{M}{}\left(\frac{{9}}{{4}}{,}{3}{,}{x}\right)\right){+}{\mathrm{_C2}}{}\left({16}{}{\left({x}{-}\frac{{1}}{{2}}\right)}^{{2}}{}{U}{}\left(\frac{{5}}{{4}}{,}{3}{,}{x}\right){+}\left({20}{}{x}{-}{5}\right){}{U}{}\left(\frac{{9}}{{4}}{,}{3}{,}{x}\right)\right)$ (1.5)

In addition to computing new solutions out of reach in previous releases, when the linear ODE is of Heun type, so it has 4 regular singularities or one of its confluent cases, and it happens to be one of the special Heun function cases that can be expressible using hypergeometric functions, both kinds of solutions are now computable. This equation is of the Heun triconfluent type

 >

The algorithms in dsolve perceive that and solve it accordingly

 > $\mathrm{dsolve}\left(\mathrm{ode__4}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{-}\frac{{2}}{{3}}{}{{x}}^{{3}}{+}{x}}{}{\mathrm{HT}}{}\left(\frac{{3}{}{{6}}^{{2}}{{3}}}}{{4}}{,}{-}\frac{{15}}{{2}}{,}{-}{{6}}^{{1}}{{3}}}{,}\frac{{x}{}{{6}}^{{2}}{{3}}}}{{3}}\right){+}{\mathrm{_C2}}{}{{ⅇ}}^{\frac{{2}}{{3}}{}{{x}}^{{3}}{-}{x}}{}{\mathrm{HT}}{}\left(\frac{{3}{}{{6}}^{{2}}{{3}}}}{{4}}{,}\frac{{15}}{{2}}{,}{-}{{6}}^{{1}}{{3}}}{,}{-}\frac{{x}{}{{6}}^{{2}}{{3}}}}{{3}}\right)$ (1.6)

By using the option of indicating the method, a solution in terms of linear combinations of Airy functions (of the 0F1)  is now also computable

 > $\mathrm{dsolve}\left(\mathrm{ode__4},\left[\mathrm{hypergeometricsols}\right]\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}\left(\left({2}{}{x}{+}{1}\right){}{\mathrm{Ai}}{\prime }{}\left({{x}}^{{2}}{-}{1}\right){+}\left({2}{}{{x}}^{{2}}{+}{x}{-}{1}\right){}{\mathrm{Ai}}{}\left({{x}}^{{2}}{-}{1}\right)\right){+}{\mathrm{_C2}}{}\left(\left({2}{}{x}{+}{1}\right){}{\mathrm{Bi}}{\prime }{}\left({{x}}^{{2}}{-}{1}\right){+}\left({2}{}{{x}}^{{2}}{+}{x}{-}{1}\right){}{\mathrm{Bi}}{}\left({{x}}^{{2}}{-}{1}\right)\right)$ (1.7)

To express this solution in terms of the more general 0F1 form you can use

 > $\mathrm{convert}\left(,\mathrm{hypergeom}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}\left(\frac{\left({2}{}{x}{+}{1}\right){}\left({-}\frac{{3}{}{{3}}^{{1}}{{6}}}{}{{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right)}^{{2}}{}{}_{{0}}{F}_{{1}}{}\left({;}\frac{{1}}{{3}}{;}{-}\frac{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}}{{9}}\right)}{{\mathrm{\pi }}}{+}{\left({{x}}^{{2}}{-}{1}\right)}^{{2}}{}{{3}}^{{1}}{{3}}}{}{}_{{0}}{F}_{{1}}{}\left({;}\frac{{5}}{{3}}{;}{-}\frac{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}}{{9}}\right)\right)}{{6}{}{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right)}{+}\left({2}{}{{x}}^{{2}}{+}{x}{-}{1}\right){}\left(\frac{{{3}}^{{1}}{{3}}}{}{}_{{0}}{F}_{{1}}{}\left({;}\frac{{2}}{{3}}{;}{-}\frac{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}}{{9}}\right)}{{3}{}{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right)}{-}\frac{\left({{x}}^{{2}}{-}{1}\right){}{{3}}^{{1}}{{6}}}{}{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right){}{}_{{0}}{F}_{{1}}{}\left({;}\frac{{4}}{{3}}{;}{-}\frac{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}}{{9}}\right)}{{2}{}{\mathrm{\pi }}}\right)\right){+}{\mathrm{_C2}}{}\left(\frac{\left({2}{}{x}{+}{1}\right){}\left(\frac{{3}{}{{3}}^{{2}}{{3}}}{}{{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right)}^{{2}}{}{}_{{0}}{F}_{{1}}{}\left({;}\frac{{1}}{{3}}{;}\frac{{\left({{x}}^{{2}}{-}{1}\right)}^{{3}}}{{9}}\right)}{{\mathrm{\pi }}}{+}{{3}}^{{5}}{{6}}}{}{\left({x}{-}{1}\right)}^{{2}}{}{\left({x}{+}{1}\right)}^{{2}}{}{}_{{0}}{F}_{{1}}{}\left({;}\frac{{5}}{{3}}{;}\frac{{\left({{x}}^{{2}}{-}{1}\right)}^{{3}}}{{9}}\right)\right){}\left({x}{-}{1}\right){}\left({x}{+}{1}\right)}{\left({6}{}{{x}}^{{2}}{-}{6}\right){}{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right)}{+}\left({2}{}{{x}}^{{2}}{+}{x}{-}{1}\right){}\left(\frac{{{3}}^{{5}}{{6}}}{}{}_{{0}}{F}_{{1}}{}\left({;}\frac{{2}}{{3}}{;}{-}\frac{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}}{{9}}\right)}{{3}{}{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right)}{+}\frac{{{3}}^{{2}}{{3}}}{}\left({{x}}^{{2}}{-}{1}\right){}{\mathrm{\Gamma }}{}\left(\frac{{2}}{{3}}\right){}{}_{{0}}{F}_{{1}}{}\left({;}\frac{{4}}{{3}}{;}{-}\frac{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}}{{9}}\right)}{{2}{}{\mathrm{\pi }}}\right)\right)$ (1.8)

A similar situation, this time with regards to this equation of the Heun biconfluent class

 >
 > $\mathrm{dsolve}\left(\mathrm{ode__5}\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{\mathrm{HB}}{}\left(\frac{{1}}{{3}}{,}\sqrt{{6}}{,}{-3}{,}\frac{{14}{}\sqrt{{6}}}{{9}}{,}\frac{\sqrt{{6}}{}{x}}{{6}}\right){}{{ⅇ}}^{{-}\frac{{x}{}\left({x}{+}{6}\right)}{{6}}}{+}\frac{{\mathrm{_C2}}{}{\mathrm{HB}}{}\left({-}\frac{{1}}{{3}}{,}\sqrt{{6}}{,}{-3}{,}\frac{{14}{}\sqrt{{6}}}{{9}}{,}\frac{\sqrt{{6}}{}{x}}{{6}}\right){}{{ⅇ}}^{{-}\frac{{x}{}\left({x}{+}{6}\right)}{{6}}}}{{{x}}^{{1}}{{3}}}}$ (1.9)

The solution is expressible in terms of linear combinations of modified Bessel functions of the 1st and 2nd kinds

 > $\mathrm{dsolve}\left(\mathrm{ode__5},\left[\mathrm{hypergeometricsols}\right]\right)$
 ${y}{}\left({x}\right){=}\frac{{\mathrm{_C1}}{}{{ⅇ}}^{{-}\frac{{x}{}\left({x}{+}{6}\right)}{{12}}}{}\left(\left({{x}}^{{2}}{+}{6}{}{x}{+}{16}\right){}{{I}}_{\frac{{1}}{{3}}}{}\left(\frac{\left({4}{+}{x}\right){}\sqrt{\left({4}{+}{x}\right){}{x}}}{{12}}\right){+}\left({4}{+}{x}\right){}\sqrt{\left({4}{+}{x}\right){}{x}}{}{{I}}_{\frac{{4}}{{3}}}{}\left(\frac{\left({4}{+}{x}\right){}\sqrt{\left({4}{+}{x}\right){}{x}}}{{12}}\right)\right)}{\sqrt{{4}{+}{x}}{}{{x}}^{{1}}{{6}}}}{+}\frac{{\mathrm{_C2}}{}{{ⅇ}}^{{-}\frac{{x}{}\left({x}{+}{6}\right)}{{12}}}{}\left(\left({{x}}^{{2}}{+}{6}{}{x}{+}{16}\right){}{{K}}_{\frac{{1}}{{3}}}{}\left(\frac{\left({4}{+}{x}\right){}\sqrt{\left({4}{+}{x}\right){}{x}}}{{12}}\right){-}\left({4}{+}{x}\right){}\sqrt{\left({4}{+}{x}\right){}{x}}{}{{K}}_{\frac{{4}}{{3}}}{}\left(\frac{\left({4}{+}{x}\right){}\sqrt{\left({4}{+}{x}\right){}{x}}}{{12}}\right)\right)}{\sqrt{{4}{+}{x}}{}{{x}}^{{1}}{{6}}}}$ (1.10)

The following example is of the same kind but more general: it belongs to the Heun general class, but again it is one of those special cases where pFq function solutions exist, in this case 2F1 with rational coefficients

 >
 > $\mathrm{dsolve}\left(\mathrm{ode__6}\right)$
 ${y}{}\left({x}\right){=}\frac{{\mathrm{_C1}}{}{\mathrm{HG}}{}\left(\frac{{-}{4}{}{I}{}\sqrt{{2}}{+}{7}}{{4}{}{I}{}\sqrt{{2}}{+}{7}}{,}\frac{{45927}{}\left({}\right)}{}\right)}{}$