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):$\displaystyle \frac{\partial ^2w}{\partial t^2}+at^b\frac{\partial^4w}{\partial x^4}=0$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Particular solutions:\\
$w_0(x,t)=1$,\\
$w_1(x,t)=x$,\\
$w_2(x,t)=x^2$,\\
$w_3(x,t)=x^3$,\\
$\displaystyle w_4(x,t)=x^4-\frac {4!a}{(b+1)(b+2)}t^{b+2}$,\\
$\displaystyle w_5(x,t)=x^5-\frac {5!a}{(b+1)(b+2)}xt^{b+2}$,\\
$\displaystyle w_6(x,t)=x^6-\frac {6!a}{2!(b+1)(b+2)}x^2t^{b+2}$,\\
$\displaystyle w_7(x,t)=x^7-\frac {7!a}{3!(b+1)(b+2)}x^3t^{b+2}$,\\
$\displaystyle w_8(x,t)=x^8-\frac {8!a}{4!(b+1)(b+2)}x^4t^{b+2}+\frac {8!a^2}{(b+1)(b+2)(2b+3)(2b+4)}t^{2b+4}$,\\
$\dots$,\\
$\displaystyle w_n(x,t)=x^n+\sum_{k=1}^{n-4k\geq0} \frac {(-1)^kn!a^kt^{kb+2k}x^{n-4k}}{(n-4k)!k!\left(\frac{b+1}{b+2}\right)_k(b+2)^{2k}}$.
\medskip

2. Particular solutions:\\
$w_0(x,t)=t$,\\
$w_1(x,t)=xt$,\\
$w_2(x,t)=x^2t$,\\
$w_3(x,t)=x^3t$,\\
$\displaystyle w_4(x,t)=x^4t-\frac {4!a}{(b+2)(b+3)}t^{b+3}$,\\
$\displaystyle w_5(x,t)=x^5t-\frac {5!a}{(b+2)(b+3)}xt^{b+3}$,\\
$\displaystyle w_6(x,t)=x^6t-\frac {6!a}{2!(b+2)(b+3)}x^2t^{b+3}$,\\
$\displaystyle w_7(x,t)=x^7t-\frac {7!a}{3!(b+2)(b+3)}x^3t^{b+3}$,\\
$\displaystyle w_8(x,t)=x^8t-\frac {8!a}{4!(b+2)(b+3)}x^4t^{b+3}+\frac {8!a^2}{(b+2)(b+3)(2b+4)(2b+5)}t^{2b+5}$,\\
$\dots$,\\
$\displaystyle w_n(x,t)=x^nt+\sum_{k=1}^{n-4k\geq0} \frac {(-1)^kn!a^kt^{kb+2k+1}x^{n-4k}}{(n-4k)!k!\left(\frac{b+3}{b+2}\right)_k(b+2)^{2k}}$.
Remarks:Here $b\neq-2$, $\displaystyle b\neq-2+\frac 1k$, $\displaystyle b\neq-2-\frac 1k$, where $k=1,\,2,\,3,\,\dots$
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latvija
City:Sigulda
Statistic information
Submission date:Fri 09 May 2008 19:22
Edits by author:1
Last edit by author:Tue 06 Oct 2015 18:42

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