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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):$\displaystyle y^2\frac{\partial ^2w}{\partial y^2}+a\frac{\partial^2w}{\partial
x^2}+bw=0$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Solutions:\hfill\break
$w_0(x,y)=\cos(x\sqrt{\frac ba})$,\\
$w_1(x,y)=y\cos(x\sqrt{\frac ba})$,\\
$w_2(x,y)=y^2\cos(x\sqrt{\frac {b+2}a})$,\\
$w_3(x,y)=y^3\cos(x\sqrt{\frac {b+6}a})$,\\
$w_4(x,y)=y^4\cos(x\sqrt{\frac {b+4\cdot3}a})$,\\
$...$,\\
$w_n(x,y)=y^n\cos(x\sqrt{\frac {b+n\cdot{(n-1)}}a})$.\\
\medskip

2. Solutions:\\
$w_0(x,y)=\sin(x\sqrt{\frac ba})$,\\
$w_1(x,y)=y\sin(x\sqrt{\frac ba})$,\\
$w_2(x,y)=y^2\sin(x\sqrt{\frac {b+2}a})$,\\
$w_3(x,y)=y^3\sin(x\sqrt{\frac {b+6}a})$,\\
$w_4(x,y)=y^4\sin(x\sqrt{\frac {b+4\cdot3}a})$,\\
$...$,\\
$w_n(x,y)=y^n\sin(x\sqrt{\frac {b+n\cdot{(n-1)}}a})$.
Novelty:New solution(s) / integral(s)
Admin's Comment:The substitution $y=ke^z$ leads to the constant coefficient equation\hfill\break
$w_{zz}-w_z+aw_{xx}}+bw=0$.
Author/Contributor's Details
Last name:Stepuchev
First name:Valeriy
Country:Latviya
City:Sigulda
Statistic information
Submission date:Tue 08 Apr 2008 19:39
Edits by author:0

Edit (Only for author/contributor)


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