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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:\\
$\displaystyle w_n(t,x)=x^n\cos(ct)+\sum^{n-mk\geq0}_{k=1}
{\frac{(-a)^kn!t^{2k}x^{n-mk}g(k,ct)}{(n-mk)!(2k)!}}$.\\

2. Particular solutions:\\
$\displaystyle w_n(t,x)=\frac{x^n\sin(ct)}c+
\sum^{n-mk\geq0}_{k=1}{\frac{(-a)^kn!t^{2k+1}x^{n-mk}
g(k+1,ct)}{(n-mk)!(2k+1)!}}$,\hfill\break
where\\
$g(k,0)=1; \ k=0,1,2,...$;\\
$g(0,ct)=\cos(ct)$,\\
$g(1,ct)=\frac {\sin(ct)}{ct}$,\\
$g(2,ct)=\frac {3\sin(ct)}{c^3t^3}-\frac {3\cos(ct)}{c^2t^2}$,\\
$g(3,ct)=\frac {45\sin(ct)}{c^5t^5}-\frac {45\cos(ct)}{c^4t^4}-
\frac {15\sin(ct)}{c^3t^3}$,\\
$...$,\\
$g(k,ct)=\frac {(2k-1)(2k-3)}{c^2t^2}[g(k-1,ct)-g(k-2,ct)]$.
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latvija
City:Sigulda
Statistic information
Submission date:Sun 12 Sep 2010 19:40
Edits by author:0

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