MiniLogo

EqWorld

The World of Mathematical Equations

IPM Logo

Exact Solutions Methods Software For Authors Math Forums

EqArchive: Add Equation/Solution > View Equation

 English only

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):$\displaystyle \frac{\partial^3w}{\partial t^3}+
ae^{bt}\frac{\partial ^mw}{\partial x^m}=0$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Particular solutions:\\
$w_0(t,x)=1$,\\
$w_1(t,x)=x$,\\
$...$,\\
$\displaystyle w_n(t,x)=x^n+\sum_{k=1}^{n-mk\geq0} 
\frac {(-a)^kn!e^{kbt}x^{n-mk}}{(n-mk)!b^{3k}(k!)^3}$.\\

2. Particular solutions:\\
$w_0(t,x)=t$,\\
$w_1(t,x)=tx$,\\
$...$,\\
$\displaystyle w_n(t,x)=tx^n+\sum_{k=1}^{n-mk\geq0}
\frac {(-a)^kn!e^{kbt}x^{n-mk}(bt-c_k)}{(n-mk)!b^{3k+1}(k!)^3}$,\\
where $c_1=3; \ c_2=\frac{9}2; \ c_3=\frac{11}2; \ ...; \ 
c_k=c_{k-1}+\frac 3k$.\\

3. Particular solutions:\\
$w_0(t,x)=t^2$,\\
$w_1(t,x)=t^2x$,\\
$...$,\\
$\displaystyle w_n(t,x)=t^2x^n+
\sum_{k=1}^{n-mk\geq0}
\frac {(-a)^kn!e^{kbt}x^{n-mk}(b^2t^2+c_kbt+s_k)}{(n-mk)!b^{3k+2}(k!)^3}$,\\
where $c_1=-6; \ ...; \ c_k=c_{k-1}-\frac 6k$;\\
$s_1=12; \ ...; \ s_k=s_{k-1}-\frac {3c_k}{k}-\frac {6}{k^2}$.
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latvija
City:Sigulda
Statistic information
Submission date:Mon 20 Sep 2010 17:20
Edits by author:0

Edit (Only for author/contributor)


The EqWorld website presents extensive information on solutions to various classes of ordinary differential equations, partial differential equations, integral equations, functional equations, and other mathematical equations.

Copyright © 2006-2011 Andrei D. Polyanin, Alexei I. Zhurov and Alexander L. Levitin