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View Equation

The database contains 327 equations (8 equations are awaiting activation).

Equation data
Category:3. Linear Partial Differential Equations
Subcategory:3.2. Second-Order Hyperbolic Equations
Equation(s):$\displaystyle \frac{\partial^2w}{\partial t^2}=b\frac{\partial ^2w}{\partial x^2}+c^2w$.
Solution(s),
Transformation(s),
Integral(s)
:
Solutions:\\
$\displaystyle w_n(x,t)=\sum^{n-2k\geq0}_{k=0}{\frac{n!b^kg(k,ct)}{(n-2k)!(2k)!}x^{n-2k}t^{2k}}$,\\
$\displaystyle w_n(x,t)=\sum^{n-2k\geq0}_{k=0}{\frac{n!b^kg(k+1,ct)}{(n-2k)!(2k+1)!}x^{n-2k}t^{2k+1}}$,\\
where $n=0,\,1,\,\dots\,$ and \\
$g(k,0)=1,$\\
$\displaystyle g(0,ct)=\cosh(ct)$,\\
$\displaystyle g(1,ct)=\frac {\sinh(ct)}{ct}$,\\
$\displaystyle g(2,ct)=-\frac {3\sinh(ct)}{c^3t^3}+\frac {3\cosh(ct)}{c^2t^2}$,\\
$\displaystyle g(3,ct)=\frac {45\sinh(ct)}{c^5t^5}-\frac {45\cosh(ct)}{c^4t^4}+\frac {15\sinh(ct)}{c^3t^3}$,\\
\dots\,,\\
$\displaystyle g(k,ct)=\frac {(2k-1)(2k-3)}{c^2t^2}[g(k-2,ct)-g(k-1,ct)]$, $k=2,\,3,\,\dots$
Remarks:For other solutions see, for example, "Handbook of Linear Partial Differential \hbox{Equations} for Engineers and Scientists" (Chapman \& Hall/CRC, 2002, p. 287) by A.~D.~Polyanin
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Middle(s) name:Germanovich
Country:Latviya
City:Sigulda
Statistic information
Submission date:Sun 16 Mar 2008 15:51
Edits by author:1
Last edit by author:Thu 28 Aug 2008 14:44

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