New solving algorithms for the entire 3F2, 2F2, 1F2 and 0F2 hypergeometric classes of equations were implemented. These new solvable classes are the ones you obtain from the standard pFq equations,

PDEtools:-declare(y(x), prime = x);

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

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

for pFq in [hypergeom([], [d,e], x), hypergeom([c], [d,e], x), hypergeom([b,c], [d,e], x), hypergeom([a,b,c], [d,e], x)]
do
    FunctionAdvisor(DE, pFq, y(x));
od;

$\left[y=\mathrm{hypergeom}\left(\left[a\,b\,c\right]\,\left[d\,e\right]\,x\right)\,\left[\mathrm{y\text{'}\text{'}\text{'}}=\frac{\left(\left(-a-b-c-3\right)x+d+e+1\right)\mathrm{y\text{'}\text{'}}}{x\left(x-1\right)}+\frac{\left(\left(\left(-b-c-1\right)a-\left(c+1\right)\left(b+1\right)\right)x+de\right)\mathrm{y\text{'}}}{x^2\left(x-1\right)}-\frac{yabc}{x^2\left(x-1\right)}\right]\right]$

by changing variables using a composition of the general transformations

x -> R(x), y -> P(x)*y;

$x\mapsto R\left(x\right),y\mapsto P\left(x\right)\cdot y$

where R(x) is a rational function and P(x) is arbitrary, and

x -> (a*x^k+b)/(c*x^k+d);

$x\mapsto \frac{a\cdot x^k+b}{c\cdot x^k+d}$

that is, a power composed with a Mobius transformation where $a,b,c,d,k$ are arbitrary constants with respect to x. Hence, the Maple system can now solve the equivalence problem to these hypergeometric equations under rational transformations and also in the case where these transformations are composed with fractional or symbolic powers. These results by our research team are new in the literature [1].

Examples

The following equation admits three pFq hypergeometric solutions computed after determining a rational transformation relating the equation to the 0F2 equation (the first one in the output of the FunctionAdvisor above)

diff(y(x), x, x, x) = - 6/x*(x-3)/(x-2)*diff(y(x), x, x) - 6/x^2*(x^2-6*x+10)/(x-2)^2*diff(y(x), x) - (x-1)*(x-2)^3/x^11*y(x);

dsolve((5));

$y=\mathrm{c\_\_1}\mathrm{hypergeom}\left(\left[\right]\,\left[\frac{1}{2}\,\frac{3}{4}\right]\,\frac{{\left(x-1\right)}^4}{64x^8}\right)+\frac{\mathrm{c\_\_2}\left(x-1\right)\mathrm{hypergeom}\left(\left[\right]\,\left[\frac{3}{4}\,\frac{5}{4}\right]\,\frac{{\left(x-1\right)}^4}{64x^8}\right)}{x^2}+\frac{\mathrm{c\_\_3}\left(x-1\right)\mathrm{hypergeom}\left(\left[\right]\,\left[\frac{5}{4}\,\frac{3}{2}\right]\,\frac{{\left(x-1\right)}^4}{64x^8}\right)\left|x-1\right|}{x^4}$

When the hypergeometric parameters are such that there are not three independent hypergeometric solutions, the new algorithm uses MeijerG functions to represent the missing ones, as in:

diff(y(x), x, x, x) = -(12*x^2-5*x+1)/x^2/(-1+2*x)*diff(y(x), x, x) + 1/x^3*(20*x^2-c-5*x+2*c*x-24*x^3)/(-1+2*x)^2*diff(y(x), x) + c*y(x)/x^4/(-1+2*x)^2;

dsolve((7));

$y=\mathrm{c\_\_1}\mathrm{hypergeom}\left(\left[c\right]\,\left[1\right]\,\frac{-1+2x}{x}\right)+\mathrm{c\_\_2}\mathrm{MeijerG}\left(\left[\left[-c+1\right]\,\left[\right]\right]\,\left[\left[0\,0\right]\,\left[\right]\right]\,\frac{-1+2x}{x}\right)+\mathrm{c\_\_3}\mathrm{MeijerG}\left(\left[\left[0\,-c+1\right]\,\left[\right]\right]\,\left[\left[0\,0\,0\right]\,\left[\right]\right]\,-\frac{-1+2x}{x}\right)$

Second order nonlinear equations appear in the mathematical formulation of problems in many areas. The methods of symmetries and integrating factors, implemented in Maple in previous releases, are the most important ones. However, large ODE classes escape these methods or are more properly solved by other means. For example the class of 2nd order nonlinear ODEs that can be obtained by reducing third order linear ODEs exploiting their scale invariance, admits explicit and tidy solutions that can only be obtained if that relationship - instead of the symmetries - is unveiled. For Maple 12, a related new algorithm was developed and implemented for solving that class, defined around the nonlinear equation

diff(y(x),x,x) = -(3*y(x) + c[2](x))*diff(y(x),x) - y(x)^3 - c[2](x)*y(x)^2 - c[1](x)*y(x) - c[0](x);

$\mathrm{y\text{'}\text{'}}=-\left(3y+c_2\left(x\right)\right)\mathrm{y\text{'}}-y^3-c_2\left(x\right)y^2-c_1\left(x\right)y-c_0\left(x\right)$

where the c[j](x) are arbitrary functions. The solution to this equation can be expressed in terms of the solutions of a third order linear equation:

dsolve((9));

$y=\frac{{\mathrm{DESol}\left(\left\{c_0\left(x\right)\mathrm{\_Y}\left(x\right)+c_1\left(x\right)\mathrm{\_Y\text{'}}+c_2\left(x\right)\mathrm{\_Y\text{'}\text{'}}+\mathrm{\_Y\text{'}\text{'}\text{'}}\right\}\,\left\{\mathrm{\_Y}\left(x\right)\right\}\right)}^{\text{'}}}{\mathrm{DESol}\left(\left\{c_0\left(x\right)\mathrm{\_Y}\left(x\right)+c_1\left(x\right)\mathrm{\_Y\text{'}}+c_2\left(x\right)\mathrm{\_Y\text{'}\text{'}}+\mathrm{\_Y\text{'}\text{'}\text{'}}\right\}\,\left\{\mathrm{\_Y}\left(x\right)\right\}\right)}$

Note the derivative of a DESol structure in the numerator of the right-hand side. This nonlinear ODE that the system can now solve generates an ODE class when it is transformed using

x -> F(x), y -> P(x)*y;

$x\mapsto F\left(x\right),y\mapsto P\left(x\right)\cdot y$

The actual family of equations that is now solvable in Maple 12, for any value of the functions F(x) and P(x), is

PDEtools:-declare((F, P)(x));

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

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

diff(y(x), x,x) = ((diff(F(x), x,x)*P(x) - 2*diff(F(x),x)*diff(P(x),x) - diff(F(x),x)^2*c[2](F(x))*P(x))/diff(F(x),x)/P(x) - 3*diff(F(x),x)*P(x)*y(x))*diff(y(x),x) - diff(F(x),x)^2*P(x)^2*y(x)^3 - diff(F(x),x)*(3*diff(P(x),x) + c[2](F(x))*P(x)*diff(F(x),x))*y(x)^2 + (diff(F(x), x,x)*diff(P(x),x) - diff(F(x),x)*diff(P(x), x,x) - diff(F(x),x)^2*c[2](F(x))*diff(P(x),x) - c[1](F(x))*P(x)*diff(F(x),x)^3)/diff(F(x),x)/P(x)*y(x) - c[0](F(x))*diff(F(x),x)^2/P(x);

$\mathrm{y\text{'}\text{'}}=\left(\frac{\mathrm{F\text{'}\text{'}}P-2\mathrm{F\text{'}}\mathrm{P\text{'}}-{\mathrm{F\text{'}}}^2{c}_2\left(F\right)P}{\mathrm{F\text{'}}P}-3\mathrm{F\text{'}}Py\right)\mathrm{y\text{'}}-{\mathrm{F\text{'}}}^2{P}^2{y}^3-\mathrm{F\text{'}}\left(3\mathrm{P\text{'}}+c_2\left(F\right)P\mathrm{F\text{'}}\right)y^2+\frac{\left(\mathrm{F\text{'}\text{'}}\mathrm{P\text{'}}-\mathrm{F\text{'}}\mathrm{P\text{'}\text{'}}-{\mathrm{F\text{'}}}^2{c}_2\left(F\right)\mathrm{P\text{'}}-c_1\left(F\right)P{\mathrm{F\text{'}}}^3\right)y}{\mathrm{F\text{'}}P}-\frac{c_0\left(F\right){\mathrm{F\text{'}}}^2}{P}$

The new algorithm recognizes whether a given equation is of this form and if so it computes the values of F(x), P(x) and the c[j](x), and with them constructs a 3rd order linear ODE equivalent to the given 2nd order nonlinear equation. This algorithm is automatically combined by dsolve with the new algorithms for computing hypergeometric solutions for 3rd order linear ODEs, resulting in new solutions available in Maple 12 for a large number of 2nd order nonlinear ODEs that were formerly considered "unsolvable".

Examples

This second order nonlinear ODE is solved by transforming it into a third order linear ODE that admits Liouvillian (trigonometric) solutions

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

dsolve((14));

The following nonlinear equation is solved the same way, but the related third order linear ODE only admits hypergeometric solutions and so is solved using the corresponding new algorithms for that problem in Maple 12

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

$\mathrm{y\text{'}\text{'}}=-3\mathrm{y\text{'}}y-y^3+\frac{1}{x}$

dsolve((16));

$y=\frac{24\mathrm{c\_\_1}{x}^2\mathrm{hypergeom}\left(\left[\right]\,\left[\frac{3}{2}\,2\right]\,\frac{x^2}{8}\right)+\mathrm{c\_\_1}{x}^4\mathrm{hypergeom}\left(\left[\right]\,\left[\frac{5}{2}\,3\right]\,\frac{x^2}{8}\right)+12\mathrm{c\_\_2}x\mathrm{hypergeom}\left(\left[\right]\,\left[\frac{1}{2}\,\frac{3}{2}\right]\,\frac{x^2}{8}\right)+4\mathrm{c\_\_2}{x}^3\mathrm{hypergeom}\left(\left[\right]\,\left[\frac{3}{2}\,\frac{5}{2}\right]\,\frac{x^2}{8}\right)-24\mathrm{MeijerG}\left(\left[\left[\right]\,\left[\right]\right]\,\left[\left[2\,\frac{1}{2}\,0\right]\,\left[\right]\right]\,-\frac{x^2}{8}\right)+24\mathrm{MeijerG}\left(\left[\left[\right]\,\left[\right]\right]\,\left[\left[1\,\frac{1}{2}\,0\right]\,\left[\right]\right]\,-\frac{x^2}{8}\right)}{12x\left(\mathrm{c\_\_1}{x}^2\mathrm{hypergeom}\left(\left[\right]\,\left[\frac{3}{2}\,2\right]\,\frac{x^2}{8}\right)+\mathrm{c\_\_2}x\mathrm{hypergeom}\left(\left[\right]\,\left[\frac{1}{2}\,\frac{3}{2}\right]\,\frac{x^2}{8}\right)+\mathrm{MeijerG}\left(\left[\left[\right]\,\left[\right]\right]\,\left[\left[1\,\frac{1}{2}\,0\right]\,\left[\right]\right]\,-\frac{x^2}{8}\right)\right)}$

A new algorithm, solving the entire Abel Inverse Linear (AIL) class of equations (see references) in terms of hypergeometric functions, was developed and implemented for Maple 12. This class of equations is the one you obtain from the standard AIL equation

diff(y(x),x) = (a[0] + a[1]*y(x) + a[2]*y(x)^2 + a[3]*y(x)^3)/((s[0] + s[1]*x)*y(x) + r[0]+r[1]*x);

under general transformations

x -> F(x), y -> (P(x)*y + Q(x)*y)/(S(x)*y + T(x)*y);

$x\mapsto F\left(x\right),y\mapsto \frac{P\cdot y+Q\left(x\right)\cdot y}{S\left(x\right)\cdot y+T\left(x\right)\cdot y}$

where $F$ is rational and $R,P,Q,S,T$ are arbitrary. The novelty with regards to previous Maple releases where this same class was partially solved is that in Maple 12 any equation obtained using a rational form of $F$ is now solved.

Examples

The following equation, involving only rational coefficients, admits a solution that involves only combinations of fractional and integer powers after all hypergeometric functions got simplified

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

$\mathrm{y\text{'}}=\frac{\left(-x+\frac{5}{2}\right)y^3}{2x{\left(-x+2\right)}^3}+\frac{\left(\frac{x}{2}-3\right)y^2}{x{\left(-x+2\right)}^2}+\frac{y}{x\left(-x+2\right)}$

dsolve((20));

$\mathrm{c\_\_1}-\frac{625\left(\left(y-\frac{2}{5}\right)x^3+\left(-y^2-\frac{22y}{5}+\frac{8}{5}\right)x^2+\frac{\left(-8+\frac{49y^2}{2}+24y\right)x}{5}-6y^2\right)\sqrt{\frac{-2x-y+4}{-8+\left(y+4\right)x}}\sqrt{6}\sqrt{10}}{54{\left(\frac{\left(2y-2\right)x-5y+4}{-8+\left(y+4\right)x}\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}x{\left(-8+\left(y+4\right)x\right)}^2}=0$

odetest((21), (20)); # verify this result - see ?odetest

$0$