Nonlinear ODEs

The Maple dsolve command now includes a new implementation of integrating factor methods for systematically determining (when they exist) integrating factors of varied types for Ordinary Differential Equations. The integrating factor method is comparable with the symmetry method: both consist of transforming the nonlinear ODE problem into a overdetermined linear PDE system, the former searching for integrating factors - the cornerstone for reductions of order - and the latter searching for symmetry generators, with which reductions of order through changes of variables can be performed. The integrating factor approach implemented in Maple 8 can also perform multiple reductions of order in one step when more than one integrating factor is found. For nonlinear ODE problems, this combination of symmetry and integrating factor methods makes Maple's ODE solver unique among its peers.

Examples:

The integrating factor approach leads to special function solutions for various nonlinear ODE families.

PDEtools[declare](y(x), prime=x);

$y\left(x\right)\mathrm{will\; now\; be\; displayed\; as}y$

$\mathrm{derivatives\; with\; respect\; to}x\mathrm{of\; functions\; of\; one\; variable\; will\; now\; be\; displayed\; with\; \text{'}}$

ode := diff(y(x), x$3) = (1/y(x)*diff(y(x),x)-2/x)*diff(y(x),x,x)+1/y(x)/x*diff(y(x),x)^2+1/x^3*y(x);

$\mathrm{ode}≔\mathrm{y\text{'}\text{'}\text{'}}=\left(\frac{\mathrm{y\text{'}}}{y}-\frac{2}{x}\right)\mathrm{y\text{'}\text{'}}+\frac{{\mathrm{y\text{'}}}^2}{yx}+\frac{y}{x^3}$

sol := dsolve(ode);

$\mathrm{sol}≔y=\mathrm{c\_\_2}\mathrm{BesselJ}\left(2\mathrm{I}\,2\mathrm{c\_\_1}\sqrt{x}\right)+\mathrm{c\_\_3}\mathrm{BesselY}\left(2\mathrm{I}\,2\mathrm{c\_\_1}\sqrt{x}\right)$

odetest( sol, ode );

$0$

An example where the linearization process implicit in this approach is more transparent:

ode := diff(y(x),x,x,x) = 13/8/x^3*y(x) - 1/8*(-20*diff(y(x),x,x)*x^2 + 26*diff(y(x),x)*x)/x^3 - 1/8*(24*diff(y(x),x)*x^3*diff(y(x),x,x) - 20*diff(y(x),x)^2*x^2)/x^3/y(x);

$\mathrm{ode}≔\mathrm{y\text{'}\text{'}\text{'}}=\frac{13y}{8x^3}-\frac{-20x^2\mathrm{y\text{'}\text{'}}+26x\mathrm{y\text{'}}}{8x^3}-\frac{24\mathrm{y\text{'}}\mathrm{y\text{'}\text{'}}{x}^3-20{\mathrm{y\text{'}}}^2{x}^2}{8x^{3}y}$

This nonlinear ODE example has three integrating factors, which are determined solving a related linear ODE. The integrating factors are then used simultaneously by the new Maple 8 ODE routines to reduce the ODE order by three in one step. To follow the solving process, you can turn ON the userinfo facility for dsolve.

infolevel[dsolve] := 4;

${\mathrm{infolevel}}_{\mathrm{dsolve}}≔4$

sol := dsolve(ode);

Methods for third order ODEs:
--- Trying classification methods ---
trying 3rd order ODE linearizable_by_differentiation
differential order: 3; trying a linearization to 4th order
trying differential order: 3; missing variables
trying differential order: 3; exact nonlinear
trying 3rd order, integrating factor of the form mu(y'') for some mu
Trying the formal computation of integrating factors depending on any 2 of [x, y, y', y'']
         *** Sublevel 2 ***
         Methods for first order ODEs:
         --- Trying classification methods ---
         trying a quadrature
         trying 1st order linear
         <- 1st order linear successful
-> Calling odsolve with the ODE, 4*diff(diff(diff(y(x),x),x),x)*x^3+10*diff(diff(y(x),x),x)*x^2-7*diff(y(x),x)*x+7*y(x), y(x)
   *** Sublevel 2 ***
   Methods for third order ODEs:
   --- Trying classification methods ---
   trying a quadrature
   checking if the LODE has constant coefficients
   checking if the LODE is of Euler type
   <- LODE of Euler type successful
Successful computation of 3 integrating factors: [y(x)*x, x^(-1/4+1/4*29^(1/2))*y(x), x^(-1/4-1/4*29^(1/2))*y(x)]
Attempting computing related first integrals...
<- high order, formal computation of integrating factors successful

$\mathrm{sol}≔y=-\frac{2\sqrt{145\mathrm{c\_\_3}{x}^{\frac{9}{4}+\frac{\sqrt{29}}{4}}\sqrt{29}-145\mathrm{c\_\_2}{x}^{\frac{9}{4}-\frac{\sqrt{29}}{4}}\sqrt{29}-841\mathrm{c\_\_3}{x}^{\frac{9}{4}+\frac{\sqrt{29}}{4}}-841\mathrm{c\_\_2}{x}^{\frac{9}{4}-\frac{\sqrt{29}}{4}}+1682\mathrm{c\_\_1}x}}{29},y=\frac{2\sqrt{145\mathrm{c\_\_3}{x}^{\frac{9}{4}+\frac{\sqrt{29}}{4}}\sqrt{29}-145\mathrm{c\_\_2}{x}^{\frac{9}{4}-\frac{\sqrt{29}}{4}}\sqrt{29}-841\mathrm{c\_\_3}{x}^{\frac{9}{4}+\frac{\sqrt{29}}{4}}-841\mathrm{c\_\_2}{x}^{\frac{9}{4}-\frac{\sqrt{29}}{4}}+1682\mathrm{c\_\_1}x}}{29}$

For second order nonlinear ODEs, the symmetry algorithms introduced in Maple 7 have been refined, now leading to simpler solutions in varied cases.

Example:

The following ODE is from Kamke's book. Two nontrivial symmetries for it are found by dsolve routines leading to a solution in terms of Jacobi elliptic functions (note the use of the new dsolve option useint).

PDEtools[declare](f(x));

$f\left(x\right)\mathrm{will\; now\; be\; displayed\; as}f$

ode := 2*diff(y(x),x,x)*y(x) - diff(y(x),x)^2 - 6*diff(f(x),x,x)/diff(f(x),x)*y(x)*diff(y(x),x) + 2*(-2*diff(f(x),x,x,x)/diff(f(x),x)+6*diff(f(x),x,x)^2/diff(f(x),x)^2)*y(x)^2-8*y(x)^3 = 0;

$\mathrm{ode}≔2y\mathrm{y\text{'}\text{'}}-{\mathrm{y\text{'}}}^2-\frac{6\mathrm{f\; \text{'}\text{'}}y\mathrm{y\text{'}}}{\mathrm{f\; \text{'}}}+2\left(-\frac{2\mathrm{f\; \text{'}\text{'}\text{'}}}{\mathrm{f\; \text{'}}}+\frac{6{\mathrm{f\; \text{'}\text{'}}}^2}{{\mathrm{f\; \text{'}}}^2}\right)y^2-8y^3=0$

sol := dsolve(ode, y(x), useint);

Methods for second order ODEs:--- Trying classification methods ---
trying 2nd order Liouville
trying 2nd order WeierstrassP
trying 2nd order JacobiSN
differential order: 2; trying a linearization to 3rd order
trying 2nd order ODE linearizable_by_differentiation
trying 2nd order, 2 integrating factors of the form mu(x,y)
trying differential order: 2; missing variables
-> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y')^2
   --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries ---
   -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y')^2
trying 2nd order, integrating factors of the form mu(x,y)/(y')^n, only the singular cases
trying symmetries linear in x and y(x)
trying differential order: 2; exact nonlinear
trying 2nd order, integrating factor of the form mu(y')
trying 2nd order, integrating factor of the form mu(x,y')
trying 2nd order, integrating factor of the form mu(x,y)/(y')^n, only the general case
trying 2nd order, integrating factor of the form mu(y,y')
trying differential order: 2; mu polynomial in y'
trying 2nd order, integrating factor of the form mu(x,y)
differential order: 2; looking for linear symmetries
-> trying 2nd order, the S-function method
   -> trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for the S-function
   -> trying 2nd order, the S-function method
   -> trying 2nd order, No Point Symmetries Class V
      --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries ---
      -> trying 2nd order, No Point Symmetries Class V
   -> trying 2nd order, No Point Symmetries Class V
      --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries ---
      -> trying 2nd order, No Point Symmetries Class V
   -> trying 2nd order, No Point Symmetries Class V
      --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries ---
      -> trying 2nd order, No Point Symmetries Class V
trying 2nd order, integrating factor of the form mu(x,y)/(y')^n, only the general case
-> trying 2nd order, dynamical_symmetries, only a reduction of order through one integrating factor of the form G(x,y)/(1+H(x,y)*y')^2
   --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries ---
   -> trying 2nd order, dynamical_symmetries, only a reduction of order through one integrating factor of the form G(x,y)/(1+H(x,y)*y')^2
--- Trying Lie symmetry methods, 2nd order ---
   -> Computing symmetries using: way = 3
   -> Computing symmetries using: way = 5

   <- successful computation of symmetries.2nd order, symmetry integrators to be used are: [can, can2, gon]
2nd order, trying integration using method can
-> Computing canonical coordinates for the symmetry [1/diff(f(x),x), 2/diff(f(x),x)^2*diff(diff(f(x),x),x)*y]
   -> Calling odsolve with the ODE, diff(y(x),x) = 2*diff(diff(f(x),x),x)/diff(f(x),x)*y(x), y(x)
      *** Sublevel 2 ***
      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear
      <- 1st order linear successful
-> Rewriting ODE in canonical coordinates by means of differential invariants
Try computing 1 more symmetries for ODE written in canonical coordinates
   -> Computing symmetries using: way = 3
   Found another symmetry:

Computing a convenient ordering to use the 2 symmetries availableTry integration with the canonical coordinates of the symmetry [1/diff(f(x),x), 2/diff(f(x),x)^2*diff(diff(f(x),x),x)*y]
-> Calling odsolve with the ODE, diff(_b(_a),_a) = -1/2*_b(_a)*(8*_b(_a)^2*_a^3+1)/_a, _b(_a), explicit, HINT = [[_a, -3/2*_b]]
   *** Sublevel 2 ***
   symmetry methods on request
   1st order, trying reduction of order with given symmetries:

   1st order, trying the canonical coordinates of the invariance group-> Computing canonical coordinates for the symmetry [_a, -3/2*_b]
      -> Calling odsolve with the ODE, diff(y(x),x) = -3/2*y(x)/x, y(x)
         *** Sublevel 3 ***
         Methods for first order ODEs:
         --- Trying classification methods ---
         trying a quadrature
         trying 1st order linear
         <- 1st order linear successful
   -> The canonical coordinates may not have unique inverse.
      Trying gauging the symmetry to the form [0, eta(x,y)]
   -> Computing canonical coordinates for the symmetry [0, 4*_a^3*_b^3-_b]
   <- 1st order, canonical coordinates successful
<- differential order: 2; canonical coordinates successful

$\left[\frac{1}{\mathrm{f\; \text{'}}}\&comma;\frac{2\mathrm{f\; \text{'}\text{'}}y}{{\mathrm{f\; \text{'}}}^2}\right]$

$\left[\frac{f}{\mathrm{f\; \text{'}}}\&comma;\frac{2yf\mathrm{f\; \text{'}\text{'}}}{{\mathrm{f\; \text{'}}}^2}-2y\right]$

$\left[\mathrm{\_a}\&comma;-\frac{3\mathrm{\_b}}{2}\right]$

$\mathrm{sol}≔y=\frac{{\mathrm{f\; \text{'}}}^2\left({\mathrm{JacobiSN}\left(\mathrm{c\_\_1}\left(-\mathrm{c\_\_2}+f\right)\&comma;\frac{\sqrt{2}}{2}\right)}^2-1\right){\mathrm{c\_\_1}}^2}{2}$

map(odetest, {sol}, ode);

$\left\{0\right\}$

New routines for solving two parameterized Abel ODE classes were added. This further enhances the top-level performance of Maple 7 dsolve in tackling ODEs of this type, that is, of the form $y\&apos;=f\left(x\right)y^3+g\left(x\right)y^2+h\left(x\right)y+j\left(x\right)$. The two integrable classes are based on recent work.

Examples:

New Abel solvable class with solutions expressed using hypergeometric 0F1 (Bessel) functions:

infolevel[dsolve] := 0:

ode := diff(y(x),x) = 1/2*(x^2-x-1)/(x+1)^2/x^2*y(x)^3-1/2*(3*x+2)/x^2/(x+1)*y(x)^2;

$\mathrm{ode}≔\mathrm{y\text{'}}=\frac{\left(x^2-x-1\right)y^3}{2{\left(x+1\right)}^2{x}^2}-\frac{\left(3x+2\right)y^2}{2x^2\left(x+1\right)}$

sol := dsolve(ode,implicit);

$\mathrm{sol}≔\mathrm{c\_\_1}+\frac{\left(-y\sqrt{5}x+\left(-y-2\right)x-2y\right)\mathrm{BesselK}\left(\frac{\sqrt{5}}{2}\&comma;-\sqrt{\frac{x^3+\left(-y^2+2y+1\right)x^2+\left(y^2+2y\right)x+y^2}{\left(x+1\right)y^2{x}^2}}\right)-2\mathrm{BesselK}\left(\frac{\sqrt{5}}{2}+1\&comma;-\sqrt{\frac{x^3+\left(-y^2+2y+1\right)x^2+\left(y^2+2y\right)x+y^2}{\left(x+1\right)y^2{x}^2}}\right)\sqrt{\frac{x^3+\left(-y^2+2y+1\right)x^2+\left(y^2+2y\right)x+y^2}{\left(x+1\right)y^2{x}^2}}yx}{\left(y\sqrt{5}x+\left(y+2\right)x+2y\right)\mathrm{BesselI}\left(\frac{\sqrt{5}}{2}\&comma;-\sqrt{\frac{x^3+\left(-y^2+2y+1\right)x^2+\left(y^2+2y\right)x+y^2}{\left(x+1\right)y^2{x}^2}}\right)-2\mathrm{BesselI}\left(\frac{\sqrt{5}}{2}+1\&comma;-\sqrt{\frac{x^3+\left(-y^2+2y+1\right)x^2+\left(y^2+2y\right)x+y^2}{\left(x+1\right)y^2{x}^2}}\right)\sqrt{\frac{x^3+\left(-y^2+2y+1\right)x^2+\left(y^2+2y\right)x+y^2}{\left(x+1\right)y^2{x}^2}}yx}=0$

odetest( sol, ode );

$0$

Another new Abel solvable class, with solutions expressed using hypergeometric  1F1 (Whittaker) functions:

ode := diff(y(x),x) = -(2*x+1)/(2*x^2-1)^2*y(x)^3+(-1+2*x+6*x^2)/(2*x^2-1)^2*y(x)^2;

$\mathrm{ode}≔\mathrm{y\text{'}}=-\frac{\left(2x+1\right)y^3}{{\left(2x^2-1\right)}^2}+\frac{\left(6x^2+2x-1\right)y^2}{{\left(2x^2-1\right)}^2}$

sol := dsolve(ode,implicit);

$\mathrm{sol}≔\mathrm{c\_\_1}+\frac{\left(\left(-8x-4\right)\sqrt{2}-8x-8\right)\mathrm{WhittakerM}\left(\frac{1}{2}+\frac{\sqrt{2}}{8}\&comma;\frac{1}{4}\&comma;\frac{\left(4x^2-2yx-y-2\right)\sqrt{2}}{4y{x}^2-2y}\right)-8\left(\left(\frac{1}{2}+x\right)\sqrt{2}-1-x\right)\mathrm{WhittakerM}\left(\frac{\sqrt{2}}{8}-\frac{1}{2}\&comma;\frac{1}{4}\&comma;\frac{\left(4x^2-2yx-y-2\right)\sqrt{2}}{4y{x}^2-2y}\right)}{\left(\left(8x+4\right)\sqrt{2}+8x+8\right)\mathrm{WhittakerW}\left(\frac{1}{2}+\frac{\sqrt{2}}{8}\&comma;\frac{1}{4}\&comma;\frac{\left(4x^2-2yx-y-2\right)\sqrt{2}}{4y{x}^2-2y}\right)+\left(1-x\sqrt{2}\right)\mathrm{WhittakerW}\left(\frac{\sqrt{2}}{8}-\frac{1}{2}\&comma;\frac{1}{4}\&comma;\frac{\left(4x^2-2yx-y-2\right)\sqrt{2}}{4y{x}^2-2y}\right)}=0$

odetest( sol, ode );

$0$

In addition, for Abel type ODEs, all routines for solving equivalence problems with respect to any of the 27 classes known by Maple dsolve can now handle fractional or abstract powers in a systematic manner (see the new Maple 8 DEtools[power_equivalent] command).

Example:

The following Abel ODE is from the suite found in Polyanin Zaitsev Handbook on ODEs (see dsolve,references).

ode := diff(y(x),x) = 1+(12*x^(7/2)+a)/x^(5/2)/y(x);

$\mathrm{ode}≔\mathrm{y\text{'}}=1+\frac{12x^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+a}{x^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}y}$

sol := dsolve(ode, implicit, useInt);

$\mathrm{sol}≔\mathrm{c\_\_1}+\frac{336\left({\int }_{\phantom{\mathrm{`\; `}}}^{\frac{\sqrt{3}\sqrt{-x^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}a}\left(4x-y\right)}{2ax}}{\left(-{\mathrm{\_a}}^2+1\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\&DifferentialD;\mathrm{\_a}\right)\sqrt{-x^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}a}+3{\left(\frac{48x^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-24x^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}y+3x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{y}^2+4a}{a}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}{2}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(48x^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-24x^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}y+3x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{y}^2+4a\right)\sqrt{3}}{\sqrt{-x^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}a}}=0$

odetest(sol,ode);

$0$

Linear ODEs

For second order linear ODEs there have been radical improvements in the computation of hypergeometric solutions. Concretely, Maple 8 dsolve can resolve the equivalence between a given ODE and one having 2F1, 1F1 or 0F1 hypergeometric solutions, whenever that equivalence involves power composed with Mobius transformations (see dsolve,hypergeometric). These algorithms new in Maple 8 largely supercede the related ones found in previous releases.

Examples

2F1 hypergeometric type of solution

infolevel[dsolve] := 4;

${\mathrm{infolevel}}_{\mathrm{dsolve}}≔4$

ode := diff(y(x),x,x) = 3/16*(x^2+10)*(4*x^6+16*x^4+23*x^2-10)/(x^2+2)^2/(x^2+5)^2/x^2*y(x);

$\mathrm{ode}≔\mathrm{y\text{'}\text{'}}=\frac{3\left(x^2+10\right)\left(4x^6+16x^4+23x^2-10\right)y}{16{\left(x^2+2\right)}^2{\left(x^2+5\right)}^2{x}^2}$

sol := dsolve(ode);

Methods for second order ODEs:
--- Trying classification methods ---
trying a quadrature
checking if the LODE has constant coefficients
checking if the LODE is of Euler type
trying a symmetry of the form [xi=0, eta=F(x)]
checking if the LODE is missing 'y'
-> Trying a Liouvillian solution using Kovacic's algorithm
<- No Liouvillian solutions exists
-> Trying a solution in terms of special functions:
   -> Bessel
   -> elliptic
   -> Legendre
   -> Whittaker
      -> hyper3: Equivalence to 1F1 under 'a power @ Moebius'
   -> hypergeometric
      -> heuristic approach
      -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under 'a power @ Moebius'
      <- hyper3 successful: received ODE is equivalent to the 2F1 ODE
      pFq equation: _a*_b*y(x)+((_b+_a+1)*x-_c)*diff(y(x),x)+(x^2-x)*diff(y(x),x,x) = 0
Transformation resolving the equivalence: {x = -3/2*t^2/(t^2+5), y(x) = 1/t^(1/4)/(t^2+5)^(1/8)/(t^2+2)^(1/2)*u(t)}
Values of hypergeometric parameters: [_a = 1/4, _b = 1/2], [_c = 3/4]

   <- hypergeometric successful
<- special function solution successful

$\mathrm{sol}≔y=\mathrm{c\_\_1}{\left(x^2+5\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}\mathrm{hypergeom}\left(\left[\frac{1}{4}\&comma;\frac{1}{2}\right]\&comma;\left[\frac{3}{4}\right]\&comma;-\frac{3x^2}{2x^2+10}\right)x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\sqrt{x^2+2}+\frac{\mathrm{c\_\_2}\mathrm{hypergeom}\left(\left[\frac{1}{2}\&comma;\frac{3}{4}\right]\&comma;\left[\frac{5}{4}\right]\&comma;-\frac{3x^2}{2x^2+10}\right)x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\sqrt{x^2+2}}{{\left(x^2+5\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}}$

odetest( sol, ode );

$0$

1F1 hypergeometric type of solution

ode := diff(y(x),x,x) = 27*x*y(x)/(2*x+1)/(x-1)^4;

$\mathrm{ode}≔\mathrm{y\text{'}\text{'}}=\frac{27xy}{\left(2x+1\right){\left(x-1\right)}^4}$

sol := dsolve(ode);

Methods for second order ODEs:
--- Trying classification methods ---
trying a quadrature
checking if the LODE has constant coefficients
checking if the LODE is of Euler type
trying a symmetry of the form [xi=0, eta=F(x)]
checking if the LODE is missing 'y'
-> Trying a Liouvillian solution using Kovacic's algorithm
<- No Liouvillian solutions exists
-> Trying a solution in terms of special functions:
   -> Bessel
   -> elliptic
   -> Legendre
   -> Whittaker
      -> hyper3: Equivalence to 1F1 under 'a power @ Moebius'
      <- hyper3 successful: received ODE is equivalent to the 1F1 ODE
   <- Whittaker successful
<- special function solution successful

$\mathrm{sol}≔y=\mathrm{c\_\_1}\mathrm{WhittakerM}\left(-\frac{1}{2}\&comma;\frac{1}{2}\&comma;\frac{4x+2}{x-1}\right)\left(x-1\right)+\mathrm{c\_\_2}\mathrm{WhittakerW}\left(-\frac{1}{2}\&comma;\frac{1}{2}\&comma;\frac{4x+2}{x-1}\right)\left(x-1\right)$

odetest( sol, ode );

$0$

convert(sol, StandardFunctions);

$y=\mathrm{c\_\_1}\mathrm{WhittakerM}\left(-\frac{1}{2}\&comma;\frac{1}{2}\&comma;\frac{4x+2}{x-1}\right)\left(x-1\right)+\mathrm{c\_\_2}\mathrm{WhittakerW}\left(-\frac{1}{2}\&comma;\frac{1}{2}\&comma;\frac{4x+2}{x-1}\right)\left(x-1\right)$

0F1 hypergeometric type of solution

ode := diff(y(x),x,x) = 1/20/x^2*(405*x^6-5670*x^4+58604*x^2+13720)/(3*x^2-14)^3*y(x);

$\mathrm{ode}≔\mathrm{y\text{'}\text{'}}=\frac{\left(405x^6-5670x^4+58604x^2+13720\right)y}{20x^2{\left(3x^2-14\right)}^3}$

sol := dsolve(ode);

Methods for second order ODEs:
--- Trying classification methods ---
trying a quadrature
checking if the LODE has constant coefficients
checking if the LODE is of Euler type
trying a symmetry of the form [xi=0, eta=F(x)]
checking if the LODE is missing 'y'
-> Trying a Liouvillian solution using Kovacic's algorithm
<- No Liouvillian solutions exists
-> Trying a solution in terms of special functions:
   -> Bessel
   -> elliptic
   -> Legendre
   -> Whittaker
      -> hyper3: Equivalence to 1F1 under 'a power @ Moebius'
   -> hypergeometric
      -> heuristic approach
      -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under 'a power @ Moebius'
      <- hyper3 successful: received ODE is equivalent to the 0F1 ODE
      pFq equation: y(x)-x*diff(y(x),x,x)-diff(y(x),x)*_c = 0
Transformation resolving the equivalence: {x = 14/5*t^2/(3*t^2-14), y(x) = 1/(3*t^3-14*t)^(1/2)*u(t)}
Values of hypergeometric parameters: [], [_c = 1]

   <- hypergeometric successful
<- special function solution successful

$\mathrm{sol}≔y=\mathrm{c\_\_1}\sqrt{3x^3-14x}\mathrm{hypergeom}\left(\left[\right]\&comma;\left[1\right]\&comma;\frac{14x^2}{15x^2-70}\right)+\mathrm{c\_\_2}\sqrt{3x^3-14x}\mathrm{BesselK}\left(0\&comma;\frac{2\sqrt{\frac{x^2}{3x^2-14}}\sqrt{70}}{5}\right)$

odetest( sol, ode );

$0$

convert(sol, Bessel);

An example with fractional power coefficients, whose solution can be expressed in elementary form (see new in Maple 8 convert,elementary):

ode := diff(y(x),x,x) = 1/4*(2/3*1/(x^(5/6))+2/x^(2/3))/x*y(x) + 1/2*(-1+x^(1/6))/x*diff(y(x),x);

$\mathrm{ode}≔\mathrm{y\text{'}\text{'}}=\frac{\left(\frac{2}{3x^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}+\frac{2}{x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right)y}{4x}+\frac{\left(-1+x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\right)\mathrm{y\text{'}}}{2x}$

sol := dsolve(ode);

Methods for second order ODEs:
--- Trying classification methods ---
trying a symmetry of the form [xi=0, eta=F(x)]
checking if the LODE is missing 'y'
-> Trying an equivalence, under non-integer power transformations,
   to LODEs admitting Liouvillian solutions.
   -> Trying a Liouvillian solution using Kovacic's algorithm
      A Liouvillian solution exists
      Reducible group (found an exponential solution)
      Reducible group (found another exponential solution)
   <- Kovacic's algorithm successful
<- Equivalence, under non-integer power transformations successful

$\mathrm{sol}≔y=\mathrm{c\_\_1}{\&ExponentialE;}^{-3x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}+\mathrm{c\_\_2}\left(81x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-18x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}+2\right){\&ExponentialE;}^{6x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}$

odetest( sol, ode );

$0$

convert(sol, elementary);

$y=\mathrm{c\_\_1}{\&ExponentialE;}^{-3x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}+\mathrm{c\_\_2}\left(81x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-18x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}+2\right){\&ExponentialE;}^{6x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}$

The Maple implementation of Kovacic's algorithm has been extended to handle fractional and abstract powers.

Example:

ode := diff(y(x),x,x) = -1/2*(x+a*x^(1/2))/x^2*diff(y(x),x)+1/4*b*(b*x+a)/x^2*y(x);

$\mathrm{ode}≔\mathrm{y\text{'}\text{'}}=-\frac{\left(x+a\sqrt{x}\right)\mathrm{y\text{'}}}{2x^2}+\frac{b\left(bx+a\right)y}{4x^2}$

dsolve(ode);

Methods for second order ODEs:
--- Trying classification methods ---
trying a symmetry of the form [xi=0, eta=F(x)]
checking if the LODE is missing 'y'
-> Trying an equivalence, under non-integer power transformations,
   to LODEs admitting Liouvillian solutions.
   -> Trying a Liouvillian solution using Kovacic's algorithm
      A Liouvillian solution exists
      Reducible group (found an exponential solution)
      Group is reducible, not completely reducible
      Solution has integrals. Trying a special function solution free of integrals...
      -> Trying a solution in terms of special functions:
         -> Bessel
         -> elliptic
         -> Legendre
         -> Whittaker
            -> hyper3: Equivalence to 1F1 under 'a power @ Moebius'
         -> hypergeometric
            -> heuristic approach
            -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under 'a power @ Moebius'
               -> trying extended approach, based on solving the ODE satisfied by the "derivative" of y(x)
                  -> ODE succesfully computed; trying to solve: diff(diff(y(x),x),x) = -4*a*(2*b*x^2+a-3*x)/x/(4*b^2*x^4+4*a*b*x^2+a^2-4*a*x)*diff(y(x),x)-1/4*y(x)*(-4*b^2*x^4-4*a*b*x^2-a^2+4*a*x)/x^4
               <- extended approach, failed in solving the ODE for the "derivative" of y(x)
         -> Mathieu
            -> Equivalence to the rational form of Mathieu ODE under 'a power @ Moebius'
               -> trying extended approach, based on solving the ODE satisfied by the "derivative" of y(x)
                  -> ODE succesfully computed; trying to solve: diff(diff(y(x),x),x) = -4*a*(2*b*x^2+a-3*x)/x/(4*b^2*x^4+4*a*b*x^2+a^2-4*a*x)*diff(y(x),x)-1/4*y(x)*(-4*b^2*x^4-4*a*b*x^2-a^2+4*a*x)/x^4
               <- extended approach, failed in solving the ODE for the "derivative" of y(x)
         -> Heun: Equivalence to the GHE or one of its 4 confluent cases under 'a power @ Moebius'
         <- Heun successful: received ODE is equivalent to the "HeunD" ODE, case "c = 0"
         Heun equation: diff(y(x),x,x)-(_a*x^4-2*x^5+4*x^3-_a-2*x)/(x-1)^3/(x+1)^3*diff(y(x),x)-(-_b*x^2+(-_c-2*_a)*x-_d)/(x-1)^3/(x+1)^3*y(x)
Transformation resolving the equivalence: {x = (2^(1/2)*(a*b)^(1/2)*t-a)/(2^(1/2)*(a*b)^(1/2)*t+a), y(x) = 1/t^(1/2)/exp(1/2*(2^(1/2)*(a*b)^(1/2)*t-a)*(2^(1/2)*(a*b)^(1/2)*t+a)/a/t)*u(t)}
Values of Heun parameters: [_a = -4*2^(1/2)*(a*b)^(1/2), _b = -1-4*2^(1/2)*(a*b)^(1/2), _c = 8*2^(1/2)*(a*b)^(1/2), _d = -4*2^(1/2)*(a*b)^(1/2)+1]

      <- Kovacic's algorithm successful
<- Equivalence, under non-integer power transformations successful

The algorithms for higher order linear ODEs have been enhanced, incorporating three new algorithms, all working in the presence of nonrational objects like arbitrary functions or fractional powers.

Examples:

ode := diff(y(x),x$3) = (-2*diff(y(x),x,x) + 2*y(x))/(-2*diff(f(x),x)^2 - 2*diff(f(x),x,x) + 2)*diff(f(x),x,x,x) + diff(y(x),x) + ((-2*diff(y(x),x,x) + 2*y(x))*diff(f(x),x)^3 + ((-6*diff(f(x),x,x) + 2)*diff(y(x),x,x) + 6*y(x)*diff(f(x),x,x) -2*y(x))*diff(f(x),x))/(-2*diff(f(x),x)^2-2*diff(f(x),x,x) + 2);

$\mathrm{ode}≔\mathrm{y\text{'}\text{'}\text{'}}=\frac{\left(-2\mathrm{y\text{'}\text{'}}+2y\right)\mathrm{f\; \text{'}\text{'}\text{'}}}{-2{\mathrm{f\; \text{'}}}^2-2\mathrm{f\; \text{'}\text{'}}+2}+\mathrm{y\text{'}}+\frac{\left(-2\mathrm{y\text{'}\text{'}}+2y\right){\mathrm{f\; \text{'}}}^3+\left(\left(-6\mathrm{f\; \text{'}\text{'}}+2\right)\mathrm{y\text{'}\text{'}}+6y\mathrm{f\; \text{'}\text{'}}-2y\right)\mathrm{f\; \text{'}}}{-2{\mathrm{f\; \text{'}}}^2-2\mathrm{f\; \text{'}\text{'}}+2}$

For arbitrary values of $f$, the following ODE has exponential solutions with simple dependence, from where a general solution can be constructed:

sol := dsolve(ode, y(x));

Methods for third order ODEs:
--- Trying classification methods ---
trying a quadrature
checking if the LODE has constant coefficients
checking if the LODE is of Euler type
trying high order exact linear fully integrable
trying to convert to a linear ODE with constant coefficients
trying differential order: 3; missing the dependent variable
-> pFq: Equivalence to the 3F2 or one of its 3 confluent cases under 'a power @ Moebius'
trying a solution in terms of MeijerG functions
-> Try computing a Rational Normal Form for the given ODE...
<- unable to resolve the Equivalence to a Rational Normal Form
checking substitution methods
checking for doubly periodic methods
trying reduction of order using simple exponentials
-> Calling odsolve with the ODE, diff(_b(_a),_a) = (diff(f(_a),_a)^3+diff(f(_a),_a)^2+3*diff(f(_a),_a)*diff(diff(f(_a),_a),_a)-diff(f(_a),_a)+diff(diff(f(_a),_a),_a)+diff(diff(diff(f(_a),_a),_a),_a)-1)*_b(_a)/(diff(f(_a),_a)^2+diff(diff(f(_a),_a),_a)-1), _b(_a)
   *** Sublevel 2 ***
   Methods for first order ODEs:
   --- Trying classification methods ---
   trying a quadrature
   trying 1st order linear
   <- 1st order linear successful
<- reduction of order using simple exponentials successful

$\mathrm{sol}≔y=\mathrm{c\_\_3}{\&ExponentialE;}^f-\frac{\mathrm{c\_\_2}{\&ExponentialE;}^{-x}}{2}+{\&ExponentialE;}^x\mathrm{c\_\_1}$

odetest( sol, ode );

$0$

In the presence of fractional or abstract power, dsolve now attempts a Rational Normal Form to remove the nonrational elements (see the new in Maple 8 DEtools[power_equivalent] command). In the displayed userinfo, you can see how the varied methods compose between themselves to result in a special function solution:

infolevel[dsolve] := 4;

${\mathrm{infolevel}}_{\mathrm{dsolve}}≔4$

ode := diff(diff(diff(y(x),x),x),x) = 1/5*(4*x*n+10*x+5-4*n)/x/(x-1)*diff(diff(y(x),x),x) - 1/25*(-9+4*a*(x-1)^2*x^(4/5) + 4*n*(x-1)*(-3+13*x) + x*(68+91*x))/(x-1)^2/x^2*diff(y(x),x) + 1/125*(4*a*(x-1)^2*(-4+9*x) + 5*x^(1/5)*(-9+4*n*(x-1)*(-3+13*x) + x*(68+91*x)))/(x-1)^3/x^(11/5)*y(x);

$\mathrm{ode}≔\mathrm{y\text{'}\text{'}\text{'}}=\frac{\left(4xn-4n+10x+5\right)\mathrm{y\text{'}\text{'}}}{5x\left(x-1\right)}-\frac{\left(-9+4a{\left(x-1\right)}^2{x}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}+4n\left(x-1\right)\left(-3+13x\right)+x\left(68+91x\right)\right)\mathrm{y\text{'}}}{25{\left(x-1\right)}^2{x}^2}+\frac{\left(4a{\left(x-1\right)}^2\left(-4+9x\right)+5x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\left(-9+4n\left(x-1\right)\left(-3+13x\right)+x\left(68+91x\right)\right)\right)y}{125{\left(x-1\right)}^3{x}^{\raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}}$

sol := dsolve(ode, y(x));

Methods for third order ODEs:
--- Trying classification methods ---
trying a quadrature
checking if the LODE has constant coefficients
checking if the LODE is of Euler type
trying high order exact linear fully integrable
trying to convert to a linear ODE with constant coefficients
trying differential order: 3; missing the dependent variable
-> pFq: Equivalence to the 3F2 or one of its 3 confluent cases under 'a power @ Moebius'
trying a solution in terms of MeijerG functions
-> Try computing a Rational Normal Form for the given ODE...
   RNF transformation found: {t = x^(4/5), u(t) = 1/x^(1/5)*y(x)*exp(-Int(1/15*(4*x*n-4*n+10*x+5)/x/(x-1),x))}
-> Calling odsolve with the ODE, diff(diff(diff(u(t),t),t),t) = ((-1/12*a*n+7/24*a)/t^2+(2/27*n^3-4/9*n^2-5/18*n+13/54)/t^3)*u(t)+(-1/4*a/t+(1/3*n^2-1/3*n+1/12)/t^2)*diff(u(t),t), u(t)
   *** Sublevel 2 ***
   Methods for third order ODEs:
   --- Trying classification methods ---
   trying a quadrature
   checking if the LODE has constant coefficients
   checking if the LODE is of Euler type
   trying high order exact linear fully integrable
   trying to convert to a linear ODE with constant coefficients
   trying differential order: 3; missing the dependent variable
   Equation is the LCLM of (1/6*(-1+2*n)/t-1/(t+2*(-1+2*n)/a))*y(t)+diff(y(t),t), 1/36*(9*a*t-8*n^2-4*n+4)/t^2*y(t)-1/6*(-1+2*n)/t*diff(y(t),t)+diff(diff(y(t),t),t)
   trying differential order: 1; missing the dependent variable
   checking if the LODE is of Euler type
   -> Attemtping a differential factorization
      trying exponential solutions
      checking if the LODE is of Euler type
      1, exponential solutions found
      exponential solutions successful
   <- differential factorization successful
      trying a quadrature
      checking if the LODE has constant coefficients
      checking if the LODE is of Euler type
      trying a symmetry of the form [xi=0, eta=F(x)]
      checking if the LODE is missing 'y'
      -> Trying a Liouvillian solution using Kovacic's algorithm
      <- No Liouvillian solutions exists
      -> Trying a solution in terms of special functions:
         -> Bessel
         <- Bessel successful
      <- special function solution successful
   <- solving the LCLM ode successful
<- solving the Rational Normal Form successful

$\mathrm{sol}≔y=\mathrm{c\_\_1}\left(x-1\right)\left(x^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}a+4n-2\right)+\mathrm{c\_\_2}\left(x-1\right)x^{\frac{1}{5}+\frac{2n}{5}}\mathrm{BesselJ}\left(\frac{\sqrt{{\left(1+2n\right)}^2}}{2}\&comma;\sqrt{a}{x}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\right)+\mathrm{c\_\_3}\left(x-1\right)x^{\frac{1}{5}+\frac{2n}{5}}\mathrm{BesselY}\left(\frac{\sqrt{{\left(1+2n\right)}^2}}{2}\&comma;\sqrt{a}{x}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\right)$

odetest( sol, ode );

$0$

Maple 8 includes enhancements and new tool commands for both linear and nonlinear differential equations

A new command, DEtools[power_equivalent], receives a linear ODE of arbitrary order - or a nonlinear first order ODE of Abel type - with non-rational coefficients involving fractional or abstract powers. It returns an equivalent - linear or Abel - ODE, with rational coefficients, followed by the transformation mapping it into the given nonrational ODE. The returned ODE is always in normal form and the coefficients' degrees in the independent variable are the lowest which could be achieved through power transformations.

A nonlinear Abel type ODE example:

PDEtools[declare](y(x), prime=x);

$y\left(x\right)\mathrm{will\; now\; be\; displayed\; as}y$

$\mathrm{derivatives\; with\; respect\; to}x\mathrm{of\; functions\; of\; one\; variable\; will\; now\; be\; displayed\; with\; \text{'}}$

ode := diff(y(x),x) = 1+(12*x^(7/2)+a)/x^(5/2)/y(x);

$\mathrm{ode}≔\mathrm{y\text{'}}=1+\frac{12x^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+a}{x^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}y}$

DEtools[power_equivalent]( ode );

A linear ODE example:

ode := diff(y(x),x,x) = -1/2*(x+a*x^(1/2))/x^2*diff(y(x),x)+1/4*b*(b*x+a)/x^2*y(x);

$\mathrm{ode}≔\mathrm{y\text{'}\text{'}}=-\frac{\left(x+a\sqrt{x}\right)\mathrm{y\text{'}}}{2x^2}+\frac{b\left(bx+a\right)y}{4x^2}$

DEtools[power_equivalent]( ode );

${\mathrm{\_b}}_{\mathrm{\_a},\mathrm{\_a}}=\left(b^2+\frac{ab}{{\mathrm{\_a}}^2}-\frac{a}{{\mathrm{\_a}}^3}+\frac{a^2}{4{\mathrm{\_a}}^4}\right)\mathrm{\_b}\left(\mathrm{\_a}\right),\left\{\mathrm{\_a}=\sqrt{x}\&comma;\mathrm{\_b}\left(\mathrm{\_a}\right)=\frac{y{\&ExponentialE;}^{-\left(\int -\frac{x+a\sqrt{x}}{4x^2}\&DifferentialD;x\right)}}{x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}\right\}$

DEtools[power_equivalent] works by resolving an Equivalence problem between the received ODE and a minimal rational one under power transformations composed with transformations of the dependent variable; that is, of the form $x->x^a,y\left(x\right)->P\left(x\right)y\left(x\right)+Q\left(x\right)$, where a is an arbitrary constant (possibly an algebraic expression) and P(x), Q(x) are arbitrary functions -- for linear ODEs $Q\left(x\right)=0$. This command is useful when tackling some typical type of nonrational ODEs using techniques for rational ones and is in use by dsolve in Maple 8 to solve varied new ODE families not solved in previous Maple releases.

The DEtools[intfactor] command can now systematically compute - when they exist - integrating factors for nonlinear ODEs (all orders) where the integrating factor can depend on as many variables {x, y, y',...} as the differential order of DE minus one. The technique explores Maple's ability to tackle PDE systems and performing differential elimination when the system contains non-polynomial coefficients (see dpolyform).

Example:

ode := diff(y(x),x,x,x) = (diff(y(x),x)*x+y(x))*diff(y(x),x,x)/(y(x)*x);

$\mathrm{ode}≔\mathrm{y\text{'}\text{'}\text{'}}=\frac{\left(x\mathrm{y\text{'}}+y\right)\mathrm{y\text{'}\text{'}}}{yx}$

DEtools[intfactor]( ode );

      Methods for first order ODEs:--- Trying classification methods ---
      trying a quadrature
      trying 1st order linear
      <- 1st order linear successful
-> Calling odsolve with the ODE, diff(y(x),x)*x+y(x), y(x)
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful
      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear
      <- 1st order linear successful
      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear
      <- 1st order linear successful

$\mathrm{The\; integrating\; factor\; is:},\frac{1}{\mathrm{\_y2}}$

$\frac{1}{xy},\frac{1}{\mathrm{y\text{'}\text{'}}}$

This new capability for computing integrating factors is in use by dsolve in Maple 8 when solving nonlinear ODEs. Both DEtools[intfactor] and dsolve can accept extra arguments allowing the user to specify a particular form of the integrating factor. This is useful when the default approach fails in solving the problem or for other reasons.

Example:

ode := diff(y(x),x,x) = (y(x) + x*diff(y(x),x) + 2*x*y(x)*diff(y(x),x) + x^2*diff(y(x),x)^2)*diff(y(x),x)/x/y(x);

$\mathrm{ode}≔\mathrm{y\text{'}\text{'}}=\frac{\left(y+x\mathrm{y\text{'}}+2y\mathrm{y\text{'}}x+{\mathrm{y\text{'}}}^2{x}^2\right)\mathrm{y\text{'}}}{xy}$

DEtools[intfactor]( ode, _mu = x*f(y,_y1) );

      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear
      <- 1st order linear successful
-> Calling odsolve with the ODE, diff(diff(y(x),x),x)*x^2-2*(diff(y(x),x)*x-y(x))*(x+1/2), y(x)
Methods for second order ODEs:
--- Trying classification methods ---
trying a quadrature
checking if the LODE has constant coefficients
checking if the LODE is of Euler type
trying a symmetry of the form [xi=0, eta=F(x)]
checking if the LODE is missing 'y'
-> Trying a Liouvillian solution using Kovacic's algorithm
   A Liouvillian solution exists
   Reducible group (found an exponential solution)
   Group is reducible, not completely reducible
<- Kovacic's algorithm successful

$\frac{xy}{{\mathrm{y\text{'}}}^2},\frac{xy{\mathrm{Ei}}_1\left(-2y\right)}{{\mathrm{y\text{'}}}^2}$

When an integrating factor is found, you can use DEtools[firint] to compute a related first integral (reduction of order; see also dsolve,education) or directly pass that integrating factor as a hint for dsolve.

Example:

ode := diff(y(x), x$3) = (1/y(x)*diff(y(x),x)-2/x)*diff(y(x),x,x)+1/y(x)/x*diff(y(x),x)^2+1/x^3*y(x);

$\mathrm{ode}≔\mathrm{y\text{'}\text{'}\text{'}}=\left(\frac{\mathrm{y\text{'}}}{y}-\frac{2}{x}\right)\mathrm{y\text{'}\text{'}}+\frac{{\mathrm{y\text{'}}}^2}{yx}+\frac{y}{x^3}$

Mu := DEtools[intfactor]( ode );

      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear
      <- 1st order linear successful

$\mathrm{{\rm M}}≔\frac{x}{y}$

dsolve(ode, _mu = Mu);

Integrating factor hint being investigated...

<- differential order: 3; exact nonlinear successful

$\mathrm{A\; reduced\; ODE\; of\; order\; 2\; is:},\frac{\mathrm{y\text{'}}}{y}+\frac{x\mathrm{y\text{'}\text{'}}}{y}+\frac{1}{x}+\mathrm{c\_\_1}$

$y=\mathrm{c\_\_2}\mathrm{BesselJ}\left(2\mathrm{I}\&comma;2\mathrm{c\_\_1}\sqrt{x}\right)+\mathrm{c\_\_3}\mathrm{BesselY}\left(2\mathrm{I}\&comma;2\mathrm{c\_\_1}\sqrt{x}\right)$

The DEtools[hypergeomsols] command can now compute hypergeometric solutions to a whole set of new linear ODE families, by resolving Equivalence problems between a given linear ODE and any of the three linear ODEs having for solutions hypergeometric 2F1, 1F1 and 0F1 functions (for details see dsolve,hypergeometric, dsolve,references).

Example:

diff(y(x),x,x) = (11/4 - x^(2*n) - 2*x^n)*n^2/(x^n+1)^2*x^(2*n-2)*y(x) + (n-1)/x*diff(y(x),x);

$\mathrm{y\text{'}\text{'}}=\frac{\left(\frac{11}{4}-x^{2n}-2x^n\right)n^2{x}^{2n-2}y}{{\left(x^n+1\right)}^2}+\frac{\left(n-1\right)\mathrm{y\text{'}}}{x}$

dsolve( (58) );

Methods for second order ODEs:
--- Trying classification methods ---
trying a symmetry of the form [xi=0, eta=F(x)]
checking if the LODE is missing 'y'
-> Trying an equivalence, under non-integer power transformations,
   to LODEs admitting Liouvillian solutions.
   -> Trying a Liouvillian solution using Kovacic's algorithm
   <- No Liouvillian solutions exists
-> Trying a solution in terms of special functions:
   -> Bessel
   -> elliptic
   -> Legendre
   -> Whittaker
      -> hyper3: Equivalence to 1F1 under 'a power @ Moebius'
      <- hyper3 successful: received ODE is equivalent to the 1F1 ODE
   <- Whittaker successful
<- special function solution successful

$y=\mathrm{c\_\_1}\mathrm{WhittakerM}\left(0\&comma;2\&comma;2\mathrm{I}\left(x^n+1\right)\right)+\mathrm{c\_\_2}\mathrm{WhittakerW}\left(0\&comma;2\&comma;2\mathrm{I}\left(x^n+1\right)\right)$

diff(y(x),x,x) = (x+1)/(x-1)^5*y(x);

$\mathrm{y\text{'}\text{'}}=\frac{\left(x+1\right)y}{{\left(x-1\right)}^5}$

dsolve( (60) );

Methods for second order ODEs:
--- Trying classification methods ---
trying a quadrature
checking if the LODE has constant coefficients
checking if the LODE is of Euler type
trying a symmetry of the form [xi=0, eta=F(x)]
checking if the LODE is missing 'y'
-> Trying a Liouvillian solution using Kovacic's algorithm
<- No Liouvillian solutions exists
-> Trying a solution in terms of special functions:
   -> Bessel
   -> elliptic
   -> Legendre
   -> Whittaker
      -> hyper3: Equivalence to 1F1 under 'a power @ Moebius'
   -> hypergeometric
      -> heuristic approach
      -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under 'a power @ Moebius'
      <- hyper3 successful: indirect Equivalence to 0F1 under ``^ @ Moebius`` is resolved
      pFq equation: y(x)-x*diff(y(x),x,x)-diff(y(x),x)*_c = 0
Transformation resolving the equivalence: {x = 1/36*(t+1)^3/(t-1)^3, y(x) = 1/(t+1)*u(t)}
Values of hypergeometric parameters: [], [_c = 4/3]

   <- hypergeometric successful
<- special function solution successful

$y=\mathrm{c\_\_1}\left(x+1\right)\mathrm{hypergeom}\left(\left[\right]\&comma;\left[\frac{4}{3}\right]\&comma;\frac{{\left(x+1\right)}^3}{36{\left(x-1\right)}^3}\right)+\mathrm{c\_\_2}\mathrm{hypergeom}\left(\left[\right]\&comma;\left[\frac{2}{3}\right]\&comma;\frac{{\left(x+1\right)}^3}{36{\left(x-1\right)}^3}\right)\left(x-1\right)$

infolevel[dsolve] := 0;

${\mathrm{infolevel}}_{\mathrm{dsolve}}≔0$

The DEtools[redode] command can now compute the ODE, admitting possibly many different integrating factors at the same time (formerly, the command could compute it admitting only one integrating factor). This command is useful for studying the inverse relation between conservation laws and Hamiltonian systems, as well as understanding the integrating factor approach as an ODE solving technique.

Example:

Consider the following three integrating factors of some (unknown) third order ODE:

mu := [y(x)*x, x^(-1/4+1/4*29^(1/2))*y(x), x^(-1/4-1/4*29^(1/2))*y(x)];

$\mathrm{\mu }≔\left[yx\&comma;x^{-\frac{1}{4}+\frac{\sqrt{29}}{4}}y\&comma;x^{-\frac{1}{4}-\frac{\sqrt{29}}{4}}y\right]$

The most general third order ODE, simultaneously admitting all these integrating factors, depends on an arbitrary function of $\mathrm{f\_\_1}\left(x\right)$ and can be computed using redode as follows:

DEtools[redode]( mu, 3, y(x) );

$\mathrm{y\text{'}\text{'}\text{'}}=\frac{13y}{8x^3}+\frac{5\mathrm{y\text{'}\text{'}}}{2x}-\frac{13\mathrm{y\text{'}}}{4x^2}+\frac{-3\mathrm{y\text{'}}\mathrm{y\text{'}\text{'}}+\frac{5{\mathrm{y\text{'}}}^2}{2x}+\mathrm{f\_\_1}\left(x\right)}{y}$

map(DEtools[mutest], mu, (64));

$\left[0\&comma;0\&comma;0\right]$

The test performed above verifies - returning zero - that the given integrating factors (the list mu) indeed turn the returned ODE exact - see mutest.

The number of given integrating factors can be different than the differential order. For example, taking just the first two integrating factors of the previous example, the most general third order ODE turned exact by them actually depends - in addition - on an arbitrary function of two variables, $\mathrm{\_F3}\left(x\&comma;y\right)$:

mu[1..2];

$\left[yx\&comma;x^{-\frac{1}{4}+\frac{\sqrt{29}}{4}}y\right]$

DEtools[redode]( (66), 3, y(x) );

$\mathrm{y\text{'}\text{'}\text{'}}={\mathrm{D}}_2\left(\mathrm{f\_\_3}\right)\left(x\&comma;y\right){\mathrm{y\text{'}}}^2+\frac{\left(16{\mathrm{D}}_1\left(\mathrm{f\_\_3}\right)\left(x\&comma;y\right)x^2+2x\mathrm{f\_\_3}\left(x\&comma;y\right)\sqrt{29}-2x\mathrm{f\_\_3}\left(x\&comma;y\right)-5\sqrt{29}+19\right)\mathrm{y\text{'}}}{8x^2}+\mathrm{f\_\_3}\left(x\&comma;y\right)\mathrm{y\text{'}\text{'}}+\frac{\mathrm{f\_\_3}\left(x\&comma;y\right){\mathrm{y\text{'}}}^2-3\mathrm{y\text{'}}\mathrm{y\text{'}\text{'}}+{\int }_{\phantom{\mathrm{`\; `}}}^y\frac{\left(8{\mathrm{f\_\_3}}_{x,x}{x}^3+2{\mathrm{f\_\_3}}_x\sqrt{29}{x}^2-2{\mathrm{f\_\_3}}_x{x}^2+5\sqrt{29}-19\right)\mathrm{\_a}}{8x^3}\&DifferentialD;\mathrm{\_a}+\mathrm{f\_\_4}\left(x\right)}{y}$

map(DEtools[mutest], mu[1..2], (67));

$\left[0\&comma;0\right]$

OFF;

The DEtools[reduceOrder] - originally written for linear ODEs - was fixed and merged into DEtools[reduce_order] (introduced in Maple 7) for nonlinear ODEs, resulting in a single command with varied and extended capabilities. This renewed routine is now one of the main Maple tools for studying the processes by means of which ODEs of order higher than one - linear and nonlinear - are solved.

The capabilities of the conversion facility for ODEs (see convert,ODEs) have been extended. It is now possible to convert to normal form ODEs of type: linear (arbitrary order), Riccati and Abel. For Abel ODEs there is a new conversion to "Rational Normal Form". These tools are useful to study the solving properties of these relevant classes of equations. Also a conversion applicable to ODEs in general, called here 'y_x', to switch the roles between the dependent (y) and independent (x) variables, has been implemented. This is a useful conversion tool in different frameworks since this transformation enters many standard and alternative interactive solving approaches.

The PDEtools[dpolyform] command, introduced in Maple 7, now admits an optional extra argument to request a nonlinear - of lower differential order - representation for nonpolynomial objects, when that is possible, instead of a linear representation of higher differential order.

New to Maple 8, Maple has the ability to numerically solve time-based linear PDE over rectangular domains. The solution procedure uses the finite difference approach to obtain these numerical solutions. There are a number of aspects of this new capability.