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\centerline{\fbox{\parbox{9.5cm}{\Large\bf UserFunction LaplaceInvariants} }}

\vspace*{3mm}\noindent{\bf LaplaceInvariants}$(u:LPDE2(C,\{z\},\{x,y\}))$
The single argument $u$ is a linear partial differential equation of the form
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$$z_{xy}+az_x+bz_y+cz=0.$$
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\noindent
The meaning of the parameters is as follows.

$C$: Coefficient type.

$z$: Dependent variable.

$x,y$: Independent variables.


\noindent{\bf Specification.}
The Laplace invariants $h$ and $k$ of $u$ are returned.


\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(z,x,y)+x*y*Df(z,x)-2*y*z=0;}

\4{\tt LaplaceInvariants deq;}

\hspace*{15mm}\fbox{\parbox{11cm}{

 The differential equation:

$$z_{xy}+xyz_x-2yz = 0$$

The Laplace Invariants


\vspace{1mm}
$$h= 3y,\2 k= 2y$$   }}

 

 

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

\4{\tt LaplaceInvariants deq;}

\hspace*{15mm}\fbox{\parbox{11cm}{

 The differential equation:

\vspace{1mm}
$$z_{xy}+\big(x+\fracsm{1}{y}\big)z_x-xz_y-3z = 0$$

The Laplace Invariants

$$h= \frac{-x^2y-x+4y}{y} ,\2 k=\frac{-x^2y-x+3y}{y}  $$   }}

 

\vspace*{3mm}
{\bf Related Functions} LaplaceTransformation, LaplaceDivisor.


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{\bf References}

E.~Darboux, {Le\c{c}ons sur la th\'{e}orie g\'{e}n\'{e}rale des surfaces}, vol II, Chapitre II, page 21-53; Chelsea Publishing Company, New York 1972.

E.~Goursat, {Le\c{c}on sur l'int\'{e}gration des
\'{e}quation aux d\'{e}riv\'{e}es partielles}, vol. I and II, A.~Hermann,
Paris 1898.

 


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