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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):$\displaystyle a\frac{\partial^2w}{\partial x^2}+y^3 \frac{\partial ^2w}{\partial y^2} =0$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Particular solutions:\\
$w_0(x,y)=1$,\\
$w_1(x,y)=x$,\\
$\displaystyle w_2(x,y)=x^2-\frac ay$,\\
$\displaystyle w_3(x,y)=x^3-\frac {3!ax}{2y}$,\\
$\displaystyle w_4(x,y)=x^4-\frac {4!ax^2}{4y}+\frac {4!a^2}{3!2!y^2}$,\\
$\dots$,\\
$\displaystyle w_n(x,y)=x^n+\sum_{k=1}^{n-2k\geq0}\frac {(-1)^kn!a^kx^{n-2k}}{k!(k+1)!(n-2k)!y^k}$.
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2. Particular solutions:\\
$w_0(x,y)=y$,\\
$w_1(x,y)=xy$,\\
$w_2(x,y)=x^2y+2a\ln y$,\\
$w_3(x,y)=x^3y+6ax\ln y$,\\
$\displaystyle w_4(x,y)=x^4y+\frac{4!ax^2}{2!}\ln y-\frac{4!a^2}{2!y}(\ln y+\frac 32)$,\\
$\dots$,\\
$\displaystyle w_n(x,y)=x^ny+\sum_{k=1}^{n-2k\geq0}\frac {(-1)^{k-1}n!a^kx^{n-2k}}{(n-2k)!y^{k-1}k!(k-1)!}(\ln y+c_k)$,\\
where $ c_1=0$, $\displaystyle c_k=c_{k-1}+\frac 1{k-1}+\frac 1k$.
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 21:02
Edits by author:0

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