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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:3.4. Other Second-Order Equations
Equation(s):$\displaystyle y\frac{\partial^2w}{\partial x^2}+\frac{\partial^2w}{\partial y^2}=0$
Solution(s),
Transformation(s),
Integral(s)
:
$1^\circ$. Solutions:\hfill\break
$$w_n(x,y)=\sum^{n-3k\ge0}_{k=0}{a_kx^{2k}y^{n-3k}},\quad n=0,\,1,\,\dots\,,$$
where $a_0=1$, $a_k=-a_{k-1}\frac{(n-3k+3)(n-3k+2)}{(2k-1)2k}$. If $\frac{n+1}3$ is an integer, then $w_n(x,y)=0$.
\medskip

$2^\circ$. Solutions:\hfill\break
$$w_{n}(x,y)=\sum^{n-3k\ge0}_{k=0}{b_kx^{2k+1}y^{n-3k}}}},\quad n=0,\,1,\,\dots\,,$
where $b_0=1$; $b_k=-b_{k-1}\frac{(n-3k+3)(n-3k+2)}{(2k+1)2k}$. If $\frac{n+1}3$ is an integer, then $w_n(x,y)=0$.
\medskip

$3^\circ$. Solutions:\hfill\break
$$w_{n}(x,y)=\sum^{n-2k\ge0}_{k=0}{a_kx^{n-2k}y^{3k}},$$
where $a_0=1;a_k=-a_{k-1}\frac{(n-2k+1)(n-2k+2)}{(3k-1)3k}$.

$4^\circ$. Solutions:\hfill\break
$$w_{n}(x,y)=\sum^{n-2k\ge0}_{k=0}{b_kx^{n-2k}y^{3k+1}},$$
where $b_0=1;b_k=-b_{k-1}\frac{(n-2k+1)(n-2k+2)}{(3k+1)3k}$.
Remarks:It is the Tricomi equation.
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valery
Middle(s) name:Germanovich
Country:Latviya
City:Sigulda
Statistic information
Submission date:Sun 02 Mar 2008 14:19
Edits by author:1
Last edit by author:Sun 16 Mar 2008 11:01

Edit (Only for author/contributor)


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