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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):$\displaystyle \biggl(\sum^m_{k=0}a_kx^k\biggr)\frac{\partial ^2w}{\partial t^2}=\frac{\partial^2w}{\partial x^2}$.
Solution(s),
Transformation(s),
Integral(s)
:
1) Particular solutions:\hfill\break
$\displaystyle w_0(x,t)=1$,\\
$w_1(x,t)=t$,\\
$...$,\\
$\displaystyle w_n(x,t)=t^n+\sum^{n-2k\geq0}_{k=1}{\frac{n!}{(n-2k)!}t^{n-2k}x^{2k}f_k(x)}$,\\
where
$\displaystyle f_k(x)=\sum^{m\cdot k}_{j=0}{b_{j,k}x^j}$,\\
$\displaystyle b_{j,1}=\frac {a_j}{(j+1)(j+2)};\quad j=0,1,...,m$,\\
$...$,\\
$\displaystyle b_{j,k}=\frac {\sum^{m_j}_{l=0}{a_lb_{j-l,k-1}}}{(2k+j-1)(2k+j)};\quad j=0,1,...,m\cdot k$,\\
if $j<m$ then $m_j=j$,\\
if $j\geq m$ then $m_j=m$.\\

2) Particular solutions:\\
$w_0(x,t)=x$,\\
$w_1(x,t)=tx$,\\
$...$,\\
$\displaystyle w_n(x,t)=t^nx+\sum^{n-2k\geq0}_{k=1}{\frac{n!}{(n-2k)!}t^{n-2k}x^{2k+1}f_k(x)}$,\\
where
$\displaystyle f_k(x)=\sum^{m\cdot k}_{j=0}{b_{j,k}x^j}$,\\
$\displaystyle b_{j,1}=\frac {a_j}{(j+2)(j+3)};\quad j=0,1,...,m$,\\
$...$,\\
$\displaystyle b_{j,k}=\frac {\sum^{m_j}_{l=0}{a_lb_{j-l,k-1}}}{(2k+j)(2k+j+1)};\quad j=0,1,...,m\cdot k$,\\
if $j<m$ then $m_j=j$,\\
if $j\geq m$ then $m_j=m$.
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latvija
City:Sigulda
Statistic information
Submission date:Thu 31 Jul 2008 20:04
Edits by author:0

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