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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):$\displaystyle ax^b \frac{\partial w}{\partial t}+\frac{\partial^3w}{\partial x^3}=0$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Particular solutions:\\
$w_0(x,t)=1$,\\
$\displaystyle w_1(x,t)=t-\frac {ax^{b+3}}{(b+1)(b+2)(b+3)}$,\\
$\displaystyle w_2(x,t)=t^2-\frac {2ax^{b+3}t}{(b+1)(b+2)(b+3)}+\frac {2a^2x^{2b+6}}{(b+1)(b+2)(b+3)(2b+4)(2b+5)(2b+6)}$,\\
$\dots$,\\
$\displaystyle w_n(x,y)=t^n+\sum_{k=1}^n\frac {(-1)^kn!a^kt^{n-k}x^{kb+3k}}{(n-k)!k!\left(\frac{b+1}{b+3}\right)_k\left(\frac{b+2}{b+3}\right)_k(b+3)^{3k}}$.
\medskip

2. Particular solutions:\\
$w_0(x,t)=x$,\\
$\displaystyle w_1(x,t)=tx-\frac {ax^{b+4}}{(b+2)(b+3)(b+4)}$,\\
$\displaystyle w_2(x,t)=t^2x-\frac {2ax^{b+4}t}{(b+2)(b+3)(b+4)}+\frac {2a^2x^{2b+7}}{(b+2)(b+3)(b+4)(2b+5)(2b+6)(2b+7)}$,\\
$\dots$,\\
$\displaystyle w_n(x,y)=t^nx+\sum_{k=1}^n\frac {(-1)^kn!a^kt^{n-k}x^{kb+3k+1}}{(n-k)!k!\left(\frac{b+2}{b+3}\right)_k\left(\frac{b+4}{b+3}\right)_k(b+3)^{3k}}$.
\medskip

3. Particular solutions:\\
$w_0(x,t)=x^2$,\\
$\displaystyle w_1(x,t)=tx^2-\frac {ax^{b+5}}{(b+3)(b+4)(b+5)}$,\\
$\displaystyle w_2(x,t)=t^2x^2-\frac {2ax^{b+5}t}{(b+3)(b+4)(b+5)}+\frac {2a^2x^{2b+8}}{(b+3)(b+4)(b+5)(2b+6)(2b+7)(2b+8)}$,\\
$\dots$,\\
$\displaystyle w_n(x,y)=t^nx^2+\sum_{k=1}^n\frac {(-1)^kn!a^kt^{n-k}x^{kb+3k+2}}{(n-k)!k!\left(\frac{b+4}{b+3}\right)_k\left(\frac{b+5}{b+3}\right)_k(b+3)^{3k}}$.
Remarks:Here, $b\neq-3$, $\displaystyle b\neq-3+\frac 2k$, $\displaystyle b\neq-3+\frac 1k$, $\displaystyle b\neq-3-\frac 1k$,
$\displaystyle b\neq-3-\frac 2k$, where $k=1,\,2,\,\dots$
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latvija
City:Sigulda
Statistic information
Submission date:Tue 13 May 2008 19:32
Edits by author:1
Last edit by author:Tue 06 Oct 2015 19:19

Edit (Only for author/contributor)


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