LaplaceTransformation

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\vspace*{3mm}\noindent{\bf LaplaceTransformation}$(u:LPDE2(C,\{z\},\{x,y\},n:NUM))$
The first 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 second argument $n$ is an integer. The meaning of the parameters is as follows.

$C$: Coefficient type.

$z$: Dependent variable.

$x,y$: Independent variables.

$n$: Integer

\noindent{\bf Specification.}
The transformed equation for $z_n$ in the notation of Darboux is 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 LaplaceTransformation(deq,1);}

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

 The differential equation:

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

The transformed equation

$$z_{xy}+\big(xy-\Frac{1}{y}\big)z_x-3yz=0$$   }}


\4{\tt LaplaceTransformation(deq,-1);}


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


 The differential equation:

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

The transformed equation


$$z_{xy}+\big(xy-\Frac{1}{y}\big)z_x-3yz=0$$   }}

 


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{\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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User Interface Functions

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