From: Jay B. <bel...@us...> - 2002-04-22 18:17:06
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Update of /cvsroot/maxima/maxima/doc/maximabook/odes In directory usw-pr-cvs1:/tmp/cvs-serv12183/maxima/doc/maximabook/odes Modified Files: odes.tex Log Message: Added a note that the linear differential equations need to have constant coefficients to be solved by desolve. Index: odes.tex =================================================================== RCS file: /cvsroot/maxima/maxima/doc/maximabook/odes/odes.tex,v retrieving revision 1.3 retrieving revision 1.4 diff -u -d -r1.3 -r1.4 --- odes.tex 19 Apr 2002 17:46:07 -0000 1.3 +++ odes.tex 22 Apr 2002 18:17:02 -0000 1.4 @@ -592,6 +592,7 @@ \subsubsection{Using \texttt{desolve}} Maxima can solve systems of linear ordinary differential equation +with constant coefficients using the \texttt{desolve} command. The differential equations must be given using functional notation, rather than with dependent variables; i.e., \texttt{diff(y(x),x)} would have to be used @@ -661,17 +662,21 @@ \subsubsection{\texttt{desolve} Method} The \texttt{desolve} routine uses the LaPlace transform to solve the -systems of differential equations, and as such works best with linear -systems. - +systems of differential equations. If $f(t)$ is defined for all $t \ge 0$, then the LaPlace transform of $f$ is given by $F(s)={\cal L}\{f(t)\} = \int_0^\infty e^{-st}f(t)\textrm{d}t$. The LaPlace transform has the useful property that a derivative is transformed into multiplication by the variable; if ${\cal L}\{f(t)\} = F(s)$, then ${\cal L}\{f'(t)\} = sF(s)-f(0)$. The LaPlace transform can thus -transform a system of differential equations into a system of ordinary -equations. If this new system can be solved, the LaPlace +transform a system of linear differential equations into a system of +ordinary equations. +(Note, however, that the LaPlace transform will transform +multiplication by a variable into differentiation; +if ${\cal L}\{f(t)\} = F(s)$, then ${\cal L}\{tf(t)\} = -F'(s)$. +The original differential equations need to have constant +coefficients to prevent this.) +If this new system can be solved, the LaPlace tranform can be inverted to give solutions of the original system of differential equations. |