#### SymmetryClass

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\vspace{5mm}\noindent{\bf SymmetryClass}$(u:ODE2(C,y,x))$
The single argument $u$ has the type $ODE2$ which abbreviates
{\em OrdinaryDifferentialEquation} of order 2, or its subtype $LODE2$ with the same parameters. Their meaning is as follows.

$C:$ Coefficient type.

$y:$ Dependent variable.

$x:$ Independent variable.

\vspace*{1mm}\noindent{\bf Specification.} The function returns the symmetry class of the submitted quasilinear second- or third-order ode as determined in Chapter 5 of the the book quoted below.

\vspace{1mm}\noindent{\bf Examples.} The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window.

\4{\tt deq:=Df(y,x,2)*y-3/4*Df(y,x)**2+y=0;}

\5{\tt SymmetryClass deq;};

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The differential Equation:

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$y''y-\fracsm{3}{4}y'^2+y=0$

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Symmetry Class ${\cal S}^2_{3,2}$ }}

\vspace{5mm}\noindent{\bf SymmetryClass}$(u:ODE3(C,y,x))$ The single argument of has the type $ODE3$ which is nonlinear third-order ode, or its subtype $LODE3$ which is a linear ode of order 3. The meaning of its parameters is as above.

\vspace*{1mm}\noindent{\bf Specification.} The function returns the symmetry class of the submitted quasilinear second- or third-order ode as determined in Chapter 5 of the the book quoted below.

\vspace{1mm}\noindent{\bf Examples.} The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window.

\4{\tt deq:=Df(y,x,3)*Df(y,x)**2+Df(y,x,3)-3*Df(y,x,2)**2*Df(y,x)=0;}

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\4{\tt SymmetryClass deq;}

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\fbox{\parbox{13cm}{
The differential Equation:

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$y'''y'^2+y'''-3y''^2y'=0$

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Symmetry Class ${\cal S}^3_6$ }}

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\4{\tt deq:=x**2*Df(y,x,3)+(x+1)*Df(y,x,2)-y=0;}

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\4{\tt SymmetryClass deq;}

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\fbox{\parbox{13cm}{
The differential Equation:

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$x^2y'''+(x+1)y''-y=0$

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Symmetry Class ${\cal S}^3_{4,5}$ }}

\vspace*{1mm}\noindent{\bf Related Functions.} Symmetries, JanetBase, DeterminingSystem.

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{\bf References}

F. Schwarz, {\em Algorithmic Lie Theory for Solving Ordinary Differential Equations}, CRC Press, 2007.

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##### User Interface Functions

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