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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):$\displaystyle \frac{\partial^2w}{\partial t^2}=a\frac{\partial ^2w}{\partial x^2}+b\frac{\partial ^2w}{\partial y^2}+c^2w$.
Solution(s),
Transformation(s),
Integral(s)
:
Particular solutions:\\
$\displaystyle w_{m,n}(t,x,y)=\sum^{m-2k\geq0}_{k=0}\sum^{n-2j\geq0}_{j=0}{\frac{(k+j)!t^{2k+2j}a^kx^{m-2k}b^jy^{n-2j}g(k+j,ct)}{k!j!(2k+2j)!(m-2k)!(n-2j)!}}$,\\
$\displaystyle w_{m,n}(t,x,y)=\sum^{m-2k\geq0}_{k=0}\sum^{n-2j\geq0}_{j=0}{\frac{(k+j)!t^{2k+2j+1}a^kx^{m-2k}b^jy^{n-2j}g(k+j+1,ct)}{k!j!(2k+2j+1)!(m-2k)!(n-2j)!}}$,\\
where $m,\,n=0,\,1,\,2,\,\dots\,$ and\\
$g(0,ct)=\cosh(ct)$,\\
$\displaystyle g(1,ct)=\frac {\sinh(ct)}{ct}$,\\
$\displaystyle g(2,ct)=-\frac {3\sinh(ct)}{c^3t^3}+\frac {3\cosh(ct)}{c^2t^2}$,\\
$\displaystyle g(3,ct)=\frac {45\sinh(ct)}{c^5t^5}-\frac {45\cosh(ct)}{c^4t^4}+\frac {15\sinh(ct)}{c^3t^3}$,\\
\dots,\\
$\displaystyle g(k,ct)=\frac {(2k-1)(2k-3)}{c^2t^2}[g(k-2,ct)-g(k-1,ct)]$.
Remarks:Here $g(k,0)=1$, $k=0,\,1,\,2,\,\dots$
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latvija
City:Sigulda
Statistic information
Submission date:Wed 21 May 2008 18:52
Edits by author:0

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