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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):$\displaystyle \frac{\partial ^3w}{\partial t^3}+
at^b\frac{\partial^mw}{\partial x^m}=0$,\hfill\break
where $b\neq-3; \ b\neq-3\pm\frac 1k; \ b\neq-3\pm\frac 2k; \ k=1,2,3,...$
Solution(s),
Transformation(s),
Integral(s)
:
1. Particular solutions:\\
$w_0(x,t)=1$,\\
$w_1(x,t)=x$,\\
$...$,\\
$\displaystyle w_n(x,t)=x^n+\sum_{k=1}^{n-mk\geq0} 
\frac {(-a)^kn!t^{bk+3k}x^{n-mk}}{(n-mk)!k!\left(\frac{b+1}{b+3}\right)_k\left(\frac{b+2}{b+3}\right)_k(b+3)^{3k}}$.\\

2. Particular solutions:\\
$w_0(x,t)=t$,\\
$w_1(x,t)=xt$,\\
$...$,\\
$\displaystyle w_n(x,t)=x^nt+\sum_{k=1}^{n-mk\geq0} 
\frac {(-a)^kn!t^{bk+3k+1}x^{n-mk}}{(n-mk)!k!\left(\frac{b+2}{b+3}\right)_k(b+3)^{3k}\left(\frac{b+4}{b+3}\right)_k}$.\\

3. Particular solutions:\\
$w_0(x,t)=t^2$,\\
$w_1(x,t)=xt^2$,\\
$...$,\\
$\displaystyle w_n(x,t)=x^nt^2+\sum_{k=1}^{n-mk\geq0} 
\frac {(-a)^kn!t^{bk+3k+2}x^{n-mk}}{(n-mk)!k!(b+3)^{3k}\left(\frac{b+4}{b+3}\right)_k\left(\frac{b+5}{b+3}\right)_k}$.
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latvija
City:Sigulda
Statistic information
Submission date:Tue 14 Sep 2010 19:37
Edits by author:1
Last edit by author:Thu 08 Oct 2015 18:44

Edit (Only for author/contributor)


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