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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:3.3. Second-Order Elliptic Equations
Equation(s):$\displaystyle a\frac{\partial^2w}{\partial x^2}+y\frac{\partial ^2w}{\partial y^2} =0$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Particular solutions:\\
$w_0(x,y)=1$,\\
$w_1(x,y)=x$,\\
$w_2(x,y)=x^2-2ay(\ln y-1)$,\\
$w_3(x,y)=x^3-6axy(\ln y-1)$,\\
$\displaystyle w_4(x,y)=x^4-\frac{4!ax^2y}{2!}(\ln y-1)+\frac{4!a^2y^2}{2!}(\ln y-\frac 52)$,\\
$\dots$,\\
$\displaystyle w_n(x,y)=x^n+\sum_{k=1}^{n-2k\geq0}\frac {(-1)^kn!a^kx^{n-2k}y^k}{(n-2k)!k!(k-1)!}(\ln y-c_k)$,\\
where $c_1=1$, $\displaystyle c_k=c_{k-1}+\frac 1{k-1}+\frac 1k$.
\medskip

2. Particular solutions:\\
$w_0(x,y)=y$,\\
$w_1(x,y)=xy$,\\
$w_2(x,y)=x^2y-ay^2$,\\
$\displaystyle w_3(x,y)=x^3y-\frac {3!axy^2}{2!}$,\\
$\dots$,\\
$\displaystyle w_n(x,y)=x^ny+\sum_{k=1}^{n-2k\geq0}\frac {(-1)^kn!a^kx^{n-2k}y^{k+1}}{k!(k+1)!(n-2k)!}$.
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latvija
City:Sigulda
Statistic information
Submission date:Thu 15 May 2008 19:34
Edits by author:0

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