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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):$\displaystyle \frac{\partial^2w}{\partial x^2}+ax^2\frac{\partial^2w}{\partial y^2}=0$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Particular solutions:\\
$w_0(x,y)=1$,\\
$w_1(x,y)=y$,\\
$\displaystyle w_2(x,y)=y^2-\frac a6x^4$,\\
$\displaystyle w_3(x,y)=y^3-\frac a2x^4y$,\\
$\displaystyle w_4(x,y)=y^4-ax^4y^2+\frac {a^2}{28}x^8$,\\
$\displaystyle w_5(x,y)=y^5-\frac {5a}3x^4y^3+\frac {5a^2}{28}x^8y$,\\
$\dots$,\\
$\displaystyle w_n(x,y)=\sum_{k=0}^{n>2k-1}b_ka^kx^{4k}y^{n-2k}$,\\
where
$b_0=1$, $\displaystyle b_1=-b_0\frac {n(n-1)}{3\cdot4},\,\dots\,, b_k=-b_{k-1}\frac {(n-2k+2)(n-2k+1)}{4k(4k-1)}$.
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2. Particular solutions:\\
$w_0(x,y)=x$,\\
$w_1(x,y)=xy$,\\
$\displaystyle w_2(x,y)=xy^2-\frac a{10}x^5$,\\
$\displaystyle w_3(x,y)=xy^3-\frac {3a}{10}x^5y$,\\
$\displaystyle w_4(x,y)=xy^4-\frac {3a}{5}x^5y^2+\frac {a^2}{60}x^9$,\\
$\displaystyle w_5(x,y)=xy^5-ax^5y^3+\frac {a^2}{12}x^9y$,\\
$\dots$,\\
$\displaystyle w_n(x,y)=\sum_{k=0}^{n>2k-1}b_ka^kx^{4k+1}y^{n-2k}$,\\
where
$b_0=1$, $\displaystyle b_1=-b_0\frac {n(n-1)}{4\cdot5},\,\dots\,, b_k=-b_{k-1}\frac {(n-2k+2)(n-2k+1)}{4k(4k+1)}$.
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latviya
City:Sigulda
Statistic information
Submission date:Tue 15 Apr 2008 21:00
Edits by author:0

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