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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:3.1. Second-Order Parabolic Equations
Equation(s):$\displaystyle  a\frac{\partial w}{\partial t}+x^3\frac{\partial^2w}{\partial x^2}=0$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Particular solutions:\hfill\break
$w_0(x,t)=1$,\\
$\displaystyle w_1(x,t)=t-\frac a{2x}$,\\
$\displaystyle w_2(x,t)=t^2-\frac {at}x+\frac {a^2}{6x^2}$,\\
$\dots$,\\
$\displaystyle w_n(x,t)=t^n+\sum_{k=1}^{n}\frac {(-1)^kn!a^kt^{n-k}}{k!(k+1)!(n-k)!x^k}$.\\
\medskip

2. Particular solutions:\\
$w_0(x,t)=x$,\\
$\displaystyle w_1(x,t)=tx+a\ln x$,\\
$\displaystyle w_2(x,t)=t^2x+2at\ln x-\frac {a^2}{2x}(\ln x+\frac 32)$,\\
$\dots$,\\
$\displaystyle w_n(x,t)=t^nx+\sum_{k=1}^{n}\frac {(-1)^{k-1}n!a^kt^{n-k}}{(n-k)!x^{k-1}k!(k-1)!}(\ln x+c_k)$,\\
where $ c_1=0$, $\displaystyle c_k=c_{k-1}+\frac 1{k-1}+\frac 1k$, and $k=2,\,3,\,\dots$
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latvija
City:Sigulda
Statistic information
Submission date:Sun 18 May 2008 20:39
Edits by author:0

Edit (Only for author/contributor)


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