Symbolics - Maple Help

Enhancements to Symbolic Capabilities in Maple 7

 Maple 7 includes a number of enhancements to its symbolic capabilities.

Ordinary and Partial Differential Equations

 Remarkable improvements and additions have been made in the sector of "Exact Solutions for Differential Equations", keeping Maple's dsolve and pdsolve commands as state-of-the-art computational tools.

PDEs

 • The Maple PDE solver, pdsolve, can now systematically tackle and in many cases successfully solve coupled systems of Partial Differential Equations.
 - There are no restrictions (but for the computer resources) to the number of equations, unknown functions, or independent variables present in the system.
 - The PDE system may also include inequations, additional Ordinary Differential Equations, and algebraic constraints involving the unknowns and the independent variables.
 - The new routines are prepared to tackle PDE systems of type linear, and polynomially nonlinear including nonpolynomial coefficients. The latter are systematically uncoupled using techniques for polynomial systems, by means of a differential extension approach (see PDEtools[dpolyform]).
 In previous releases, Maple became famous for its high performance in finding "Exact Solutions for Differential Equations". The new ability to find exact solutions to PDE systems as the ones described makes Maple 7 unique among all Computer Algebra Systems. For some examples of the type of problems which can now be tackled, see the Examples section of pdsolve,system.

ODEs

 Nonlinear ODEs
 Summary: A new implementation of symmetry methods, for systematically determining, when they exist, point symmetries for Ordinary Differential Equations, was added to Maple's dsolve. For nonlinear problems, this turns the already sophisticated Maple's ODE solver into the most advanced one among those available in computer algebra systems.
 • A new full implementation of symmetry methods using a formal approach was prepared and incorporated into Maple's dsolve. The ODE point symmetries are computed by formally uncoupling and solving the underlying linear Partial Differential Equation system. These symmetries are then used to integrate the ODE using standard methods.
 The Maple implementation of this systematic approach works with polynomially nonlinear as well as with nonpolynomial ODEs, and also in the presence of arbitrary parameters or functions. The new symmetry routines are in use by default in dsolve and can also be invoked directly by using the appropriate optional arguments (see dsolve,Lie and DEtools[symgen]). Examples:
 > infolevel[symgen] := 2:      # turn ON userinfo for symmetries
 > PDEtools[declare]( y(x), prime=x );
 ${y}{}\left({x}\right){}{\mathrm{will now be displayed as}}{}{y}$
 ${\mathrm{derivatives with respect to}}{}{x}{}{\mathrm{of functions of one variable will now be displayed with \text{'}}}$ (1)
 > diff(y(x),x,x) = -3*diff(y(x),x)^2/y(x)-2*diff(y(x),x)/x+1/(x^6*diff(y(x),x)*y(x)^6);
 ${\mathrm{y\text{'}\text{'}}}{=}{-}\frac{{3}{}{{\mathrm{y\text{'}}}}^{{2}}}{{y}}{-}\frac{{2}{}{\mathrm{y\text{'}}}}{{x}}{+}\frac{{1}}{{{x}}^{{6}}{}{\mathrm{y\text{'}}}{}{{y}}^{{6}}}$ (2)
 A 3-dimensional symmetry group is found and used to build an implicit solution as follows:
 > dsolve((2), Lie, way=formal, implicit);
 -> Computing symmetries using: way = formal
 <- successful computation of symmetries.
 $\left[{0}{,}\frac{{1}}{{{y}}^{{3}}}\right]{,}\left[{{x}}^{{2}}{,}{0}\right]{,}\left[{x}{,}{-}\frac{{3}{}{y}}{{8}}\right]$
 $\frac{{{y}}^{{4}}}{{4}}{-}\frac{\left({\mathrm{_C1}}{}{x}{-}{2}\right){}\sqrt{{\mathrm{_C1}}{}{{x}}^{{2}}{-}{2}{}{x}}}{{3}{}{{x}}^{{2}}}{-}{\mathrm{_C2}}{=}{0}{,}\frac{{{y}}^{{4}}}{{4}}{+}\frac{\left({\mathrm{_C1}}{}{x}{-}{2}\right){}\sqrt{{\mathrm{_C1}}{}{{x}}^{{2}}{-}{2}{}{x}}}{{3}{}{{x}}^{{2}}}{-}{\mathrm{_C2}}{=}{0}$ (3)
 An example from Kamke's book, parametrized by an arbitrary function $f\left(x\right)$, where only the new symmetry routines succeed in solving the problem:
 > diff(y(x),x,x) = (3*diff(y(x),x)^2*f(x) - 3*y(x)*diff(y(x),x)*diff(f(x),x) + y(x)^2*diff(f(x),x,x) - 6*y(x)^5)/(f(x)*y(x));
 ${\mathrm{y\text{'}\text{'}}}{=}\frac{{3}{}{{\mathrm{y\text{'}}}}^{{2}}{}{f}{}\left({x}\right){-}{3}{}{y}{}{\mathrm{y\text{'}}}{}{\mathrm{f\text{'}}}{+}{{y}}^{{2}}{}{\mathrm{f\text{'}\text{'}}}{-}{6}{}{{y}}^{{5}}}{{f}{}\left({x}\right){}{y}}$ (4)
 > dsolve((4), y(x), implicit);
 -> Computing symmetries using: way = 3-> Computing symmetries using: way = 5    -> Computing symmetries using: way = formal
 <- successful computation of symmetries.
 $\left[\frac{{1}}{{f}{}\left({x}\right)}{,}\frac{{y}{}{\mathrm{f\text{'}}}}{{{f}{}\left({x}\right)}^{{2}}}\right]{,}\left[\frac{{\int }{-}{f}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{{f}{}\left({x}\right)}{,}\frac{{y}{}\left({2}{}{{f}{}\left({x}\right)}^{{2}}{+}{3}{}{\mathrm{f\text{'}}}{}\left({\int }{-}{f}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\right)\right)}{}\right]$