Important efficiency improvement in the methods for first order ODEs of Abel type: The dsolve command's computational time for all parameterized Abel ODE classes is reduced almost 40 % compared with previous releases. This is achieved using new specialized subroutines for computing resultants using modular arithmetics. In addition, some families of Abel equations in second kind format, which can be transformed into Riccati equations, are handled in 1/10 or less of the previous computational time.

Example

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

$\frac{\ⅆ}{\ⅆx}y\left(x\right)=-\frac{y\left(x\right)}{y\left(x\right)+x\left(-x+a-1\right)}$

dsolve((1), y(x));

$\mathrm{c\_\_1}+\frac{x\mathrm{BesselJ}\left(a-1\,2\sqrt{-y\left(x\right)}\right)-\mathrm{BesselJ}\left(a\,2\sqrt{-y\left(x\right)}\right)\sqrt{-y\left(x\right)}}{x\mathrm{BesselY}\left(a-1\,2\sqrt{-y\left(x\right)}\right)-\mathrm{BesselY}\left(a\,2\sqrt{-y\left(x\right)}\right)\sqrt{-y\left(x\right)}}=0$

A new solving method for a Riccati family depending on two arbitrary functions, which explores different forms of the symmetries of first order Riccati equations, is available.

Example

The Riccati first order ODE family:

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

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

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

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

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

diff(y(x),x) = G(x)*y(x)^2 - (diff(F(x),x)+2*G(x)*F(x)^2)/F(x)*y(x) + (2*F(x)*diff(F(x),x)+G(x)*F(x)^3)/F(x);

$\mathrm{y\text{'}}=G{y}^2-\frac{\left(\mathrm{F\text{'}}+2G{F}^2\right)y}{F}+\frac{2F\mathrm{F\text{'}}+G{F}^3}{F}$

Independent of the values of the functions $\left\{F\,G\right\}$, a solution is now always computable as:

dsolve((4), y(x));

$y=\frac{\left(\int \frac{G}{F}\ⅆx\right)F^2+\mathrm{c\_\_1}{F}^2-1}{F\left(\int \frac{G}{F}\ⅆx+\mathrm{c\_\_1}\right)}$

The new method extends the solving capabilities for second order linear ODEs with non-rational coefficients. Independent of the value of $G$, a solution for the following ODE is now always computable.

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

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

sol := dsolve((6), y(x));

$\mathrm{sol}≔y={\ⅇ}^{\int -\frac{\left(\left(\int G{\ⅇ}^{-x}\ⅆx\right){\ⅇ}^{6x}+\mathrm{c\_\_1}{\ⅇ}^{6x}-{\ⅇ}^{4x}\right)G{\ⅇ}^{-5x}}{\int G{\ⅇ}^{-x}\ⅆx+\mathrm{c\_\_1}}\ⅆx}\mathrm{c\_\_2}$

The computation of the integrals depends on the particular value of $G$. ODE solutions can be tested by entering odetest(sol, ode). For more information, see odetest.

A new algorithm was implemented for computing Mathieu function solutions for second order linear ODEs by resolving an equivalence similar to the one formulated in previous Maple releases to compute hypergeometric 2F1, 1F1, 0F1 solutions.

Example

The equation number 2.268 from Kamke's book:

ode := 2*x*(x-1)*diff(y(x),x,x)+(2*x-1)*diff(y(x),x)+(mu*x+nu)*y(x) = 0;

$\mathrm{ode}≔2x\left(x-1\right)\mathrm{y\text{'}\text{'}}+\left(2x-1\right)\mathrm{y\text{'}}+\left(\mathrm{\mu }x+\mathrm{\nu }\right)y=0$

dsolve(ode);

$y=\mathrm{c\_\_1}\mathrm{MathieuC}\left(-\mathrm{\mu }-2\mathrm{\nu }\,\frac{\mathrm{\mu }}{2}\,\arccos \left(\sqrt{x}\right)\right)+\mathrm{c\_\_2}\mathrm{MathieuS}\left(-\mathrm{\mu }-2\mathrm{\nu }\,\frac{\mathrm{\mu }}{2}\,\arccos \left(\sqrt{x}\right)\right)$

For ODEs that admit 2F1 hypergeometric solutions, when possible, the solutions are now systematically expressed using special functions instead of uncomputed integrals. This is achieved exploring the automorphisms of the 2F1 equation. For more information, see dsolve,references.

Example

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

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

dsolve(ode);             # New output free of uncomputed integrals

Computing solutions for initial value ODE problems in more difficult cases:

Example

ode := diff(y(x),x) = 1/(x-y(x)); # Abel type ODE

$\mathrm{ode}≔\mathrm{y\text{'}}=\frac{1}{x-y}$

sol := dsolve({ode, y(0) = 1});   # correct branch for LambertW

$\mathrm{sol}≔y=-\mathrm{LambertW}\left(\mathrm{-1}\,-2{\ⅇ}^{-2+x}\right)-1+x$

simplify( eval(sol, x = 0) );     # verifying initial condition

$y\left(0\right)=1$

Note also that, new in Maple 9.5, odetest can test that dsolve's output also satisfies the initial conditions of the problem.

odetest( sol, [ode, y(0) = 1]);   # verifies 'sol' solves the ode and satisfies y(0) = 1

$\left[0\,0\right]$

Example

ode := diff(diff(y(x),x),x) + piecewise(x < 1/2, 1, 1/2 <= x, 0) * y(x) = 0;

$\mathrm{ode}≔\mathrm{y\text{'}\text{'}}+\left(\left\{\begin{array}{cc}1& x<\frac{1}{2}\\ 0& \frac{1}{2}\le x\end{array}\right.\right)y=0$

initial_values := y(0)=1, D(y)(0)=1;

$\mathrm{initial\_values}≔y\left(0\right)=1,\mathrm{D}\left(y\right)\left(0\right)=1$

dsolve( {ode, initial_values} );

$y=\left\{\begin{array}{cc}\sin \left(x\right)+\cos \left(x\right)& x<\frac{1}{2}\\ \cos \left(\frac{1}{2}\right)x+\frac{3\sin \left(\frac{1}{2}\right)}{2}+\frac{\cos \left(\frac{1}{2}\right)}{2}-\sin \left(\frac{1}{2}\right)x& \frac{1}{2}\le x\end{array}\right.$

For linear problems, like the previous one, when possible, the new routines return a solution that is continuous at the point where the piecewise function changes branches (in this example, at x = 1/2).

The DEtools[polysols] command was rewritten to compute polynomial solutions for nonlinear ODEs and systems of them.

Example

This is equation 14 from the section on nonlinear higher order ODEs in Kamke's book.

ode := diff(y(x),x)*diff(y(x),x,x,x,x) - diff(y(x),x,x)*diff(y(x),x,x,x) + diff(y(x),x)^3*diff(y(x),x,x,x) = 0;

$\mathrm{ode}≔\mathrm{y\text{'}}\mathrm{y\text{'}\text{'}\text{'}\text{'}}-\mathrm{y\text{'}\text{'}}\mathrm{y\text{'}\text{'}\text{'}}+{\mathrm{y\text{'}}}^3\mathrm{y\text{'}\text{'}\text{'}}=0$

The general solution to this equation is not known. Attempting its computation using $\mathrm{dsolve}\left(\mathrm{ode}\right)$, we obtain a partial solution, a reduction of order from 4 to 1. It is in difficult problems like this one that the new routines for computing polynomial solutions can play a crucial role. This equation, in fact, admits polynomial solutions.

ans := DEtools[polysols]( ode );

$\mathrm{ans}≔\left[y=\mathrm{\_C5}{x}^2+\mathrm{\_C4}x+\mathrm{\_C3}\right]$

Although this solution, depending only on three arbitrary constants, is not the most general for the given fourth order equation, it still has an important level of generality and is the most general solution presently known.

Example

An ODE problem with initial conditions

ode := [diff(y(x),x)=sin(x-y(x)), y(0) = 8];

$\mathrm{ode}≔\left[\mathrm{y\text{'}}=\sin \left(x-y\right)\&comma;y\left(0\right)=8\right]$

sol := dsolve(ode);     # new solution in Maple 9.5

$\mathrm{sol}≔y=x-2\arctan \left(\frac{\tan \left(4\right)x-2\tan \left(4\right)+x}{\tan \left(4\right)x+x+2}\right)+2\mathrm{\pi }$

odetest( sol, ode );    # verifies 'sol' solves the ode and satisfies y(0) = 8

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

The diff_table command is basically the inverse facility of PDEtools[declare]: it permits entering (input) expressions and their derivatives using compact mathematical notation, without using macros or aliases. The notation implemented by diff_table is the jet notation also used by the diffalg package and greatly reduces redundant typing.

Example

Let U and V be the "differentiation tables" of $u\left(x\&comma;y\&comma;t\right)$ and $v\left(x\&comma;y\&comma;t\right)$ - that is, a handy representations for these objects and their derivatives.

U := DEtools[diff_table]( u(x,y,t) ):

V := DEtools[diff_table]( v(x,y,t) ):

You can now input the functions $u\left(x\&comma;y\&comma;t\right)$ or $v\left(x\&comma;y\&comma;t\right)$ or any of its partial derivatives using mathematical notation directly, resulting in the expected expression on output.

e1 := U[y,t] + V[x,x] + U[x]*U[y] + U[]*U[x,y];

$\mathrm{e1}≔u\left(x\&comma;y\&comma;t\right)\left(\frac{{\partial }^2}{\partial x\partial y}u\left(x\&comma;y\&comma;t\right)\right)+\left(\frac{\partial }{\partial x}u\left(x\&comma;y\&comma;t\right)\right)\left(\frac{\partial }{\partial y}u\left(x\&comma;y\&comma;t\right)\right)+\frac{{\partial }^2}{\partial t\partial y}u\left(x\&comma;y\&comma;t\right)+\frac{{\partial }^2}{\partial x^2}v\left(x\&comma;y\&comma;t\right)$

diff_table can be used simultaneously with PDEtools[declare] so that both input and display are simplified while the contents of the expressions generated are as expected.

The DEtools[polysols] command was rewritten to compute polynomial solutions for nonlinear ODEs and systems of them and also for linear ODE systems. The new functionality uses the LinearFunctionalSystems package, in the case of linear ODEs, and the subroutines of the new PDEtools[TWSolutions] in the case of nonlinear ODEs. For advanced users, this new DEtools[polysols] includes programmer entry points for the different types of problems it can handle.

Examples

A linear ODE system:

sys := [diff(y1(x), x) - y2(x), diff(y2(x), x) - y3(x) - y4(x), diff(y3(x), x) - y5(x) = 0,
        diff(y4(x), x) - 2*y1(x) - 2*x*y2(x) - y5(x) = 0,
        diff(y5(x), x) - x^2*y1(x) - 2*x*y3(x) - y6(x) = 0,
        diff(y6(x), x) - x^2*y2(x) + 2*y3(x) = 0];

$\mathrm{sys}≔\left[\frac{\&DifferentialD;}{\&DifferentialD;x}\mathrm{y1}\left(x\right)-\mathrm{y2}\left(x\right)\&comma;\frac{\&DifferentialD;}{\&DifferentialD;x}\mathrm{y2}\left(x\right)-\mathrm{y3}\left(x\right)-\mathrm{y4}\left(x\right)\&comma;\frac{\&DifferentialD;}{\&DifferentialD;x}\mathrm{y3}\left(x\right)-\mathrm{y5}\left(x\right)=0\&comma;\frac{\&DifferentialD;}{\&DifferentialD;x}\mathrm{y4}\left(x\right)-2\mathrm{y1}\left(x\right)-2x\mathrm{y2}\left(x\right)-\mathrm{y5}\left(x\right)=0\&comma;\frac{\&DifferentialD;}{\&DifferentialD;x}\mathrm{y5}\left(x\right)-x^2\mathrm{y1}\left(x\right)-2x\mathrm{y3}\left(x\right)-\mathrm{y6}\left(x\right)=0\&comma;\frac{\&DifferentialD;}{\&DifferentialD;x}\mathrm{y6}\left(x\right)-x^2\mathrm{y2}\left(x\right)+2\mathrm{y3}\left(x\right)=0\right]$

In addition to the new functionality for computing solutions for ODE systems, a new option, output = solution, causes the polysols command to return output in the same closed form format used by the dsolve command, instead of a list of the solution basis for the linear system. Also, similar to the Maple dsolve command, the dependent variables of the system do not need to be specified unless really necessary.

DEtools[polysols](sys, output = solution);

Warning, received no explicit list of solving variables. The output corresponds to the ordered list [y1(x), y2(x), y3(x), y4(x), y5(x), y6(x)]

$\left[\left\{\mathrm{y1}\left(x\right)=\mathrm{\_C3}x+\mathrm{\_C2}\&comma;\mathrm{y2}\left(x\right)=\mathrm{\_C3}\&comma;\mathrm{y3}\left(x\right)=-\mathrm{\_C3}{x}^2-\mathrm{\_C2}x+\mathrm{\_C1}\&comma;\mathrm{y4}\left(x\right)=\mathrm{\_C3}{x}^2+\mathrm{\_C2}x-\mathrm{\_C1}\&comma;\mathrm{y5}\left(x\right)=-2\mathrm{\_C3}x-\mathrm{\_C2}\&comma;\mathrm{y6}\left(x\right)=\mathrm{\_C3}{x}^3+\mathrm{\_C2}{x}^2-2\mathrm{\_C1}x-2\mathrm{\_C3}\right\}\right]$

A nonlinear ODE of seventh order depending on three parameters $\left\{a\&comma;b\&comma;c\right\}$:

U := DEtools[diff_table]( u(t) ):

ODE := ((-a*t^2+a)*U[]-1/2159 * (13185*a^3*c*t^2-4335*a^3*c + 315*a^3*c*t^6-525*a^3*c*t^4+259080*a^5*t^4 - 259080*a^5*t^2+34544*a^5+2159*b) * (t+1)*(t-1))*U[t]-5/4318*a^3*(t+1)^4 * (t-1)^4*(105*t^2*c+17272*a^2-35*c)*t*U[t$4] - 1/17272*a^3*(t+1)^5*(t-1)^5*(315*t^2*c+17272*a^2-35*c)*U[t$5] - 15/2159*a^3*(63*t^4*c-70*t^2*c+34544*a^2*t^2+879*c - 17272*a^2)*t*(t+1)^2*(t-1)^2*U[t,t] - 5/4318*a^3*(315*t^4*c+103632*a^2*t^2-210*t^2*c+879*c-17272*a^2)*(t+1)^3*(t-1)^3*U[t$3] - 21/17272*c*a^3*(-1+t^2)^6*t*U[t$6]-1/34544*c*a^3*(-1+t^2)^7*U[t$7] = 0;

$\mathrm{ODE}≔\left(\left(-a{t}^2+a\right)u\left(t\right)-\frac{\left(315a^{3}c{t}^6+259080a^5{t}^4-525a^{3}c{t}^4-259080a^5{t}^2+13185a^{3}c{t}^2+34544a^5-4335a^{3}c+2159b\right)\left(t+1\right)\left(t-1\right)}{2159}\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;t}u\left(t\right)\right)-\frac{5a^3{\left(t+1\right)}^4{\left(t-1\right)}^4\left(105t^{2}c+17272a^2-35c\right)t\left(\frac{{\&DifferentialD;}^4}{\&DifferentialD;t^4}u\left(t\right)\right)}{4318}-\frac{a^3{\left(t+1\right)}^5{\left(t-1\right)}^5\left(315t^{2}c+17272a^2-35c\right)\left(\frac{{\&DifferentialD;}^5}{\&DifferentialD;t^5}u\left(t\right)\right)}{17272}-\frac{15a^3\left(63t^{4}c+34544a^2{t}^2-70t^{2}c-17272a^2+879c\right)t{\left(t+1\right)}^2{\left(t-1\right)}^2\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;t^2}u\left(t\right)\right)}{2159}-\frac{5a^3\left(315t^{4}c+103632a^2{t}^2-210t^{2}c-17272a^2+879c\right){\left(t+1\right)}^3{\left(t-1\right)}^3\left(\frac{{\&DifferentialD;}^3}{\&DifferentialD;t^3}u\left(t\right)\right)}{4318}-\frac{21c{a}^3{\left(t^2-1\right)}^{6}t\left(\frac{{\&DifferentialD;}^6}{\&DifferentialD;t^6}u\left(t\right)\right)}{17272}-\frac{c{a}^3{\left(t^2-1\right)}^7\left(\frac{{\&DifferentialD;}^7}{\&DifferentialD;t^7}u\left(t\right)\right)}{34544}=0$

In addition to the new functionality for solving nonlinear ODEs and systems of them, a new option, parameters = {...}, causes the polynomial solutions to be computed taking into account all possible values of the parameters (in this example $\left\{a\&comma;b\&comma;c\right\}$) which could lead to more solutions. In other words, the parameters become solving variables too. The approach uses differential algebra techniques.

DEtools[polysols](ODE, parameters={a,b,c});

$\left[\left\{a=a\&comma;b=b\&comma;c=c\&comma;u\left(t\right)=\mathrm{\_C3}\right\}\&comma;\left\{a=a\&comma;b=-\frac{10952}{5}{a}^5-\mathrm{\_C3}a\&comma;c=-\frac{2159a^2}{25}\&comma;u\left(t\right)=\frac{8316}{5}{a}^4{t}^6-\frac{33264}{5}{a}^4{t}^4+8316a^4{t}^2+\mathrm{\_C3}\right\}\&comma;\left\{a=0\&comma;b=0\&comma;c=c\&comma;u\left(t\right)=\mathrm{\_C9}{t}^6+\mathrm{\_C8}{t}^5+\mathrm{\_C7}{t}^4+\mathrm{\_C6}{t}^3+\mathrm{\_C5}{t}^2+\mathrm{\_C4}t+\mathrm{\_C3}\right\}\right]$

For definitions and more information, see DEtools[polysols].

A new command, DEtools[FunctionDecomposition], is available to perform rational function decomposition on overdetermined systems with parameters. The new command efficiently solves this problem, which appears frequently when solving differential equations using an equivalence approach.

Example

with(DEtools, FunctionDecomposition):

Consider the following rational system with three equations.

sys :=
[1/4*(9*C-15*F(x)*C+15*C^2*F(x)^3+2-4*C*F(x)^2+2*C^2*F(x)^4)^3/C^3/(1-
F(x)+C*F(x)^3)^5 =
9/4*x^3*(4+30*x-12*x^2-45*x^3+36*x^4)^3/(-3*x^2+2+3*x^3)^5,
C*(4*C^3*F(x)^7-4*F(x)+135*C+60*C^3*F(x)^6+315*C^2*F(x)^3-315*F(x)*C-
60*C^2*F(x)^4-12*C^2*F(x)^5-60*C*F(x)^2+60+12*C*F(x)^3)*(1-F(x)+C*F(x)^
3)/(9*C-15*F(x)*C+15*C^2*F(x)^3+2-4*C*F(x)^2+2*C^2*F(x)^4)^2 =
1/3*(-3*x^2+2+3*x^3)*(16-72*x^2+1998*x^4+240*x-360*x^3-540*x^5-2889*x^
6+2025*x^7)/x^2/(4+30*x-12*x^2-45*x^3+36*x^4)^2,
1/4*(4*C^3*F(x)^7-4*F(x)+135*C+60*C^3*F(x)^6+315*C^2*F(x)^3-315*F(x)*C
-60*C^2*F(x)^4-12*C^2*F(x)^5-60*C*F(x)^2+60+12*C*F(x)^3)*(9*C-15*F(x)*
C+15*C^2*F(x)^3+2-4*C*F(x)^2+2*C^2*F(x)^4)/C^2/(1-F(x)+C*F(x)^3)^4 =
3/4*(16-72*x^2+1998*x^4+240*x-360*x^3-540*x^5-2889*x^6+2025*x^7)*x*(4+
30*x-12*x^2-45*x^3+36*x^4)/(-3*x^2+2+3*x^3)^4];

$\mathrm{sys}≔\left[\frac{{\left(9C-15F\left(x\right)C+15C^2{F\left(x\right)}^3+2-4C{F\left(x\right)}^2+2C^2{F\left(x\right)}^4\right)}^3}{4C^3{\left(1-F\left(x\right)+C{F\left(x\right)}^3\right)}^5}=\frac{9x^3{\left(36x^4-45x^3-12x^2+30x+4\right)}^3}{4{\left(3x^3-3x^2+2\right)}^5}\&comma;\frac{C\left(4C^3{F\left(x\right)}^7-4F\left(x\right)+135C+60C^3{F\left(x\right)}^6+315C^2{F\left(x\right)}^3-315F\left(x\right)C-60C^2{F\left(x\right)}^4-12C^2{F\left(x\right)}^5-60C{F\left(x\right)}^2+60+12C{F\left(x\right)}^3\right)\left(1-F\left(x\right)+C{F\left(x\right)}^3\right)}{{\left(9C-15F\left(x\right)C+15C^2{F\left(x\right)}^3+2-4C{F\left(x\right)}^2+2C^2{F\left(x\right)}^4\right)}^2}=\frac{\left(3x^3-3x^2+2\right)\left(2025x^7-2889x^6-540x^5+1998x^4-360x^3-72x^2+240x+16\right)}{3x^2{\left(36x^4-45x^3-12x^2+30x+4\right)}^2}\&comma;\frac{\left(4C^3{F\left(x\right)}^7-4F\left(x\right)+135C+60C^3{F\left(x\right)}^6+315C^2{F\left(x\right)}^3-315F\left(x\right)C-60C^2{F\left(x\right)}^4-12C^2{F\left(x\right)}^5-60C{F\left(x\right)}^2+60+12C{F\left(x\right)}^3\right)\left(9C-15F\left(x\right)C+15C^2{F\left(x\right)}^3+2-4C{F\left(x\right)}^2+2C^2{F\left(x\right)}^4\right)}{4C^2{\left(1-F\left(x\right)+C{F\left(x\right)}^3\right)}^4}=\frac{3\left(2025x^7-2889x^6-540x^5+1998x^4-360x^3-72x^2+240x+16\right)x\left(36x^4-45x^3-12x^2+30x+4\right)}{4{\left(3x^3-3x^2+2\right)}^4}\right]$