The idea in the symmetry & ODE patterns is to match a given first order ODE to one of a set of ODE families which are invariant under different symmetry groups (to determine these invariant ODE families see equinv). These routines are an implementation of the algorithms presented in "Symmetries and first order ODE patterns, Computer Physics Communications 113 (1998) 239 (see dsolve's references).

We define the infinitesimals xi(x,y) and eta(x,y) as the coefficients of the infinitesimal symmetry generator:

f -> xi*diff(f,x) + eta*diff(f,y);

$f\→\mathrm{\ξ}\left(\frac{\partial }{\partial x}f\right)\+\mathrm{\η}\left(\frac{\partial }{\partial y}f\right)$

where x and y represent, respectively, the independent and dependent variables. The implementation in dsolve for matching first order invariant ODE families can handle the nine symmetry patterns mentioned in that CPC article:

X1 := [xi = 0, eta = F(x)*G(y)]:

X2 := [xi = F(x)*G(y), eta = 0]:

X3 := [xi = 0, eta = F(x)+G(y)]:

X4 := [xi = F(x)+G(y), eta = 0]:

X5 := [xi = G(y), eta = J(y)]:

X6 := [xi = F(x), eta = H(x)]:

X7 := [xi = F(x), eta = G(y)]:

X8 := [xi = G(y), eta = F(x)]:

X9 := [xi = a*x + b*y + c, eta = e*x + f*y + g]:

where the patterns X2, X4, X6 can be obtained from X1, X3, X5 by interchanging the roles between dependent and independent variable (x <-> y).

The explicit form of the invariant ODE families associated to the symmetry patterns 1 to 8 can be obtained using equinv. For instance the families invariant under X1, X3, X5 are given by

ODE1 := collect( DEtools[equinv](X1, y(x)), G, expand);

$\mathrm{ODE1}≔\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)=\left(\frac{\left(\frac{\&DifferentialD;}{\&DifferentialD;x}F\left(x\right)\right)\left({\int }_{\phantom{\mathrm{`\; `}}}^{y\left(x\right)}\frac{1}{G\left(\mathrm{\_a}\right)}\&DifferentialD;\mathrm{\_a}\right)}{F\left(x\right)}+\mathrm{f\_\_1}\left(x\right)\right)G\left(y\left(x\right)\right)$

ODE3 := DEtools[equinv](X3, y(x));

$\mathrm{ODE3}≔\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)=\left({\int }_{\phantom{\mathrm{`\; `}}}^{y\left(x\right)}\frac{\frac{\&DifferentialD;}{\&DifferentialD;x}F\left(x\right)}{{\left(F\left(x\right)+G\left(\mathrm{\_a}\right)\right)}^2}\&DifferentialD;\mathrm{\_a}+\mathrm{f\_\_1}\left(x\right)\right)\left(F\left(x\right)+G\left(y\left(x\right)\right)\right)$

ODE5 := DEtools[equinv](X5, y(x));

$\mathrm{ODE5}≔\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)=\frac{J\left(y\left(x\right)\right)}{G\left(y\left(x\right)\right)+\mathrm{f\_\_1}\left(-\left({\int }_{\phantom{\mathrm{`\; `}}}^{y\left(x\right)}\frac{G\left(\mathrm{\_a}\right)}{J\left(\mathrm{\_a}\right)}\&DifferentialD;\mathrm{\_a}\right)+x\right)}$

As seen in above, these families depend on the arbitrary functions {F, G, J, _F1}, and the implementation of these methods means that these ODE families - as well as those associated to the patterns X2, X4, X6, X7, X8 and X9 - can be systematically solved, in principle, for any particular form of {F, G, J, _F1}

$\mathrm{infolevel}\left[\mathrm{dsolve}\right]≔2$

${\mathrm{infolevel}}_{\mathrm{dsolve}}≔2$

$\mathrm{ode}\left[1\right]≔\mathrm{diff}\left(y\left(x\right)\&comma;x\right)=-\frac{{\left(y\left(x\right)+b\right)}^2}{\left(x+a\right)\left(1+\sin \left(y\left(x\right)\right){\left(y\left(x\right)+b\right)}^2\left(x+a\right)\right)}$

${\mathrm{ode}}_1≔\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)=-\frac{{\left(y\left(x\right)+b\right)}^2}{\left(x+a\right)\left(1+\sin \left(y\left(x\right)\right){\left(y\left(x\right)+b\right)}^2\left(x+a\right)\right)}$

$\mathrm{DEtools}\left[\mathrm{symgen}\right]\left(\mathrm{ode}\left[1\right]\&comma;\mathrm{way}=\mathrm{patterns}\right)$

   -> Computing symmetries using: way = patterns

$\left[\mathrm{\_\&xi;}={\&ExponentialE;}^{-\frac{1}{y+b}}{\left(x+a\right)}^2\&comma;\mathrm{\_\&eta;}=0\right]$

$\mathrm{collect}\left(\mathrm{dsolve}\left(\mathrm{ode}\left[1\right]\&comma;\mathrm{way}=\mathrm{patterns}\&comma;\mathrm{implicit}\right)\&comma;\mathrm{Intat}\right)$

symmetry methods on request--- Trying Lie symmetry methods, 1st order ---
   -> Computing symmetries using: way = patterns

   <- successful computation of symmetries.1st order, trying the canonical coordinates of the invariance group
<- 1st order, canonical coordinates successful

$\left[{\&ExponentialE;}^{-\frac{1}{y+b}}{\left(x+a\right)}^2\&comma;0\right]$

$-\frac{{\&ExponentialE;}^{\frac{1}{y\left(x\right)+b}}}{x+a}+{\int }_{\phantom{\mathrm{`\; `}}}^{y\left(x\right)}{\&ExponentialE;}^{\frac{1}{\mathrm{\_a}+b}}\sin \left(\mathrm{\_a}\right)\&DifferentialD;\mathrm{\_a}-\mathrm{c\_\_1}=0$

$\mathrm{odetest}\left(\&comma;\mathrm{ode}\left[1\right]\right)$

$0$

$\mathrm{ode}\left[2\right]≔\mathrm{diff}\left(y\left(x\right)\&comma;x\right)=3\left(\frac{{y\left(x\right)}^2}{{y\left(x\right)}^2}+\frac{x^2}{{y\left(x\right)}^2}\right)\arctan \left(\frac{1}{x}y\left(x\right)\right)-\frac{-1+2y\left(x\right)}{x}-\frac{-1+3y\left(x\right)}{{y\left(x\right)}^2}x$

${\mathrm{ode}}_2≔\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)=3\left(1+\frac{x^2}{{y\left(x\right)}^2}\right)\arctan \left(\frac{y\left(x\right)}{x}\right)-\frac{-1+2y\left(x\right)}{x}-\frac{\left(-1+3y\left(x\right)\right)x}{{y\left(x\right)}^2}$

$\mathrm{dsolve}\left(\mathrm{ode}\left[2\right]\right)$

Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
differential order: 1; looking for linear symmetries
trying exact
Looking for potential symmetries
trying inverse_Riccati
trying an equivalence to an Abel ODE
differential order: 1; trying a linearization to 2nd order
--- trying a change of variables {x -> y(x), y(x) -> x}
differential order: 1; trying a linearization to 2nd order
trying 1st order ODE linearizable_by_differentiation
--- Trying Lie symmetry methods, 1st order ---
   -> Computing symmetries using: way = 3
   -> Computing symmetries using: way = 4
   -> Computing symmetries using: way = 5
trying symmetry patterns for 1st order ODEs
-> trying a symmetry pattern of the form [F(x)*G(y), 0]
-> trying a symmetry pattern of the form [0, F(x)*G(y)]
-> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)]
-> trying a symmetry pattern of the form [F(x),G(x)]
-> trying a symmetry pattern of the form [F(y),G(y)]
-> trying a symmetry pattern of the form [F(x)+G(y), 0]
-> trying a symmetry pattern of the form [0, F(x)+G(y)]
<- symmetry pattern of the form [0, F(x)+G(y)] successful

$y\left(x\right)=\tan \left(\mathrm{RootOf}\left(-2x^3\tan \left(\mathrm{\_Z}\right)+2\mathrm{\_Z}{x}^3+x^2+\mathrm{c\_\_1}\right)\right)x$

$\mathrm{ode}\left[3\right]≔\mathrm{diff}\left(y\left(x\right)\&comma;x\right)-x^{a-1}{y\left(x\right)}^{1-b}H\left(\frac{x^a}{a}+\frac{{y\left(x\right)}^b}{b}\right)=0$

${\mathrm{ode}}_3≔\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)-x^{a-1}{y\left(x\right)}^{1-b}H\left(\frac{x^a}{a}+\frac{{y\left(x\right)}^b}{b}\right)=0$

$\mathrm{dsolve}\left(\mathrm{ode}\left[3\right]\right)$

Methods for first order ODEs:
--- Trying classification methods ---
trying homogeneous types:
differential order: 1; looking for linear symmetries
trying exact
Looking for potential symmetries
trying an equivalence to an Abel ODE
trying 1st order ODE linearizable_by_differentiation
--- Trying Lie symmetry methods, 1st order ---
   -> Computing symmetries using: way = 3
   -> Computing symmetries using: way = 4
   -> Computing symmetries using: way = 5
trying symmetry patterns for 1st order ODEs
-> trying a symmetry pattern of the form [F(x)*G(y), 0]
-> trying a symmetry pattern of the form [0, F(x)*G(y)]
-> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)]
1st order, trying the canonical coordinates of the invariance group
   -> Calling odsolve with the ODE, diff(y(x),x) = -y(x)/(y(x)^b)*x^a/x, y(x)
      *** Sublevel 2 ***
      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear
      trying Bernoulli
      <- Bernoulli successful
<- 1st order, canonical coordinates successful
<- symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] successful

$y\left(x\right)={\left(-\frac{x^{a}b-\mathrm{RootOf}\left(\left({\int }_{\phantom{\mathrm{`\; `}}}^{\mathrm{\_Z}}\frac{1}{-{\left({\left(-b+\mathrm{\_a}\right)}^{\frac{1}{b}}\right)}^{-b}H\left(\frac{{\left(a^{\frac{1}{a}}\right)}^{a}b+{\left({\left(-b+\mathrm{\_a}\right)}^{\frac{1}{b}}\right)}^{b}a}{ab}\right){\left(a^{\frac{1}{a}}\right)}^{a}b+{\left({\left(-b+\mathrm{\_a}\right)}^{\frac{1}{b}}\right)}^{-b}H\left(\frac{{\left(a^{\frac{1}{a}}\right)}^{a}b+{\left({\left(-b+\mathrm{\_a}\right)}^{\frac{1}{b}}\right)}^{b}a}{ab}\right){\left(a^{\frac{1}{a}}\right)}^a\mathrm{\_a}+a}\&DifferentialD;\mathrm{\_a}\right)a^2+\mathrm{c\_\_1}ab-x^{a}b\right)a}{a}\right)}^{\frac{1}{b}}$

$\mathrm{ode}\left[4\right]≔{\left(x\mathrm{diff}\left(y\left(x\right)\&comma;x\right)+y\left(x\right)+2x\right)}^2-4xy\left(x\right)-4x^2-4a=0$

${\mathrm{ode}}_4≔{\left(\left(\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\right)x+y\left(x\right)+2x\right)}^2-4xy\left(x\right)-4x^2-4a=0$

$\left[\mathrm{dsolve}\left(\mathrm{ode}\left[4\right]\&comma;\mathrm{way}=\mathrm{patterns}\right)\right]$

Methods for first order ODEs:*** Sublevel 2 ***
   symmetry methods on request
      -> Computing symmetries using: way = patterns

      <- successful computation of symmetries.1st order, trying reduction of order with given symmetries:

   1st order, trying the canonical coordinates of the invariance group-> Calling odsolve with the ODE, diff(y(x),x) = (-2*x-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
   <- 1st order, canonical coordinates successful
   1st order, trying reduction of order with given symmetries:

   1st order, trying the canonical coordinates of the invariance group<- 1st order, canonical coordinates successful

$\left[1\&comma;-2-\frac{y}{x}\right]$

$\left[1\&comma;-2-\frac{y}{x}\right]$

$\left[1\&comma;-2-\frac{y}{x}\right]$

$\left[y\left(x\right)=-\frac{x^2+a}{x}\&comma;y\left(x\right)=\frac{{\mathrm{c\_\_1}}^2-2\mathrm{c\_\_1}x-a}{x}\right]$