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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:3.5. Higher-Order Equations
Equation(s):$\displaystyle \frac{\partial^2w}{\partial t^2}+
a\frac{\partial ^mw}{\partial x^m}-c^2w=0$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Particular solutions:\\
$\displaystyle w_n(t,x)=x^n\cosh(ct)+
\sum^{n-mk\geq0}_{k=1}{\frac{(-a)^kn!t^{2k}x^{n-mk}g(k,ct)}{(n-mk)!(2k)!}}$.\\

2.Solution:\\
$\displaystyle w_n(t,x)=\frac{x^n\sinh(ct)}c+
\sum^{n-mk\geq0}_{k=1}{\frac{(-a)^kn!t^{2k+1}x^{n-mk}g(k+1,ct)}{(n-mk)!(2k+1)!}}$,\\
where $n=0,1,2,...;$\\
$g(k,0)=1;\ k=0,1,2,...$,\\
$g(0,ct)=\cosh(ct)$,\\
$g(1,ct)=\frac {\sinh(ct)}{ct}$,\\
$g(2,ct)=-\frac {3\sinh(ct)}{c^3t^3}+\frac {3\cosh(ct)}{c^2t^2}$,\\
$g(3,ct)=\frac {45\sinh(ct)}{c^5t^5}-\frac {45\cosh(ct)}{c^4t^4}+
\frac {15\sinh(ct)}{c^3t^3}$,\\
$...$,\\
$g(k,ct)=\frac {(2k-1)(2k-3)}{c^2t^2}[g(k-2,ct)-g(k-1,ct)]$.
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latvija
City:Sigulda
Statistic information
Submission date:Fri 17 Sep 2010 20:44
Edits by author:0

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