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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):$\displaystyle e^{ax}\frac{\partial ^2w}{\partial y^2}+\frac{\partial^2w}{\partial x^2}=0$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Particular solutions:\hfill\break
$w_0(x,y)=1$,\\
$w_1(x,y)=y$,\\
$\displaystyle w_2(x,y)=y^2-\frac {2e^{ax}}{a^2}$,\\
$\displaystyle w_3(x,y)=y^3-\frac {6e^{ax}y}{a^2}$,\\
$\displaystyle w_4(x,y)=y^4-\frac {12e^{ax}y^2}{a^2}+\frac {6e^{2ax}}{a^4}$,\\
$\displaystyle w_5(x,y)=y^5-\frac {20e^{ax}y^3}{a^2}+\frac {30e^{2ax}y}{a^4}$,\\
$\dots$,\\
$\displaystyle w_n(x,y)=y^n+\sum_{k=1}^{n-2k\geq0} \frac {n!(-1)^ke^{kax}y^{n-2k}}{(n-2k)!a^{2k}(k!)^2}$.\\
\medskip

2. Particular solutions:\hfill\break
$w_0(x,y)=x$,\\
$w_1(x,y)=xy$,\\
$\displaystyle w_2(x,y)=xy^2-\frac {2e^{ax}(ax-2)}{a^3}$,\\
$\displaystyle w_3(x,y)=xy^3-\frac {6e^{ax}y(ax-2)}{a^3}$,\\
$\displaystyle w_4(x,y)=xy^4-\frac {12e^{ax}y^2(ax-2)}{a^3}+\frac {6e^{2ax}(ax-3)}{a^5}$,\\
$\displaystyle w_5(x,y)=xy^5-\frac {20e^{ax}y^3(ax-2)}{a^3}+\frac {30e^{2ax}y(ax-3)}{a^5}$,\\
$\dots$,\\
$\displaystyle w_n(x,y)=xy^n+\sum_{k=1}^{n-2k\geq0}\frac {n!(-1)^ke^{kax}y^{n-2k}(ax-b_k)}{(n-2k)!a^{2k+1}(k!)^2}$,\\
where $b_1=2$, $b_2=3$, $\displaystyle b_3=\frac{11}3$, \dots, $\displaystyle b_k=b_{k-1}+\frac 2k$.
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latviya
City:Sigulda
Statistic information
Submission date:Sat 19 Apr 2008 19:25
Edits by author:0

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