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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):$\displaystyle \frac{\partial^2w}{\partial t^2}+
ae^{bt}\frac{\partial ^mw}{\partial x^m}=0$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Particular solutions:\\
$w_0(t,x)=1$,\\
$w_1(t,x)=x$,\\
$w_2(t,x)=x^2$,\\
$...$,\\
$\displaystyle w_n(t,x)=x^n+\sum_{k=1}^{n-mk\geq0} 
\frac {(-a)^kn!e^{kbt}x^{n-mk}}{(n-mk)!b^{2k}(k!)^2}$.\\

2. Particular solutions:\\
$w_0(t,x)=t$,\\
$w_1(t,x)=tx$,\\
$w_2(t,x)=tx^2$,\\
$...$,\\
$\displaystyle w_n(t,x)=tx^n+\sum_{k=1}^{n-mk\geq0}
\frac {(-a)^kn!e^{kbt}x^{n-mk}(bt-c_k)}{(n-mk)!b^{2k+1}(k!)^2}
$,\\
where $c_1=2; \ c_2=3; \ c_3=\frac{11}3; \ ...; \ c_k=c_{k-1}+\frac 2k$.
Novelty:New equation(s) & solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latvija
City:Sigulda
Statistic information
Submission date:Thu 09 Sep 2010 19:04
Edits by author:0

Edit (Only for author/contributor)


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