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View Equation

The database contains 327 equations (8 equations are awaiting activation).

Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.4. Other Second-Order Equations
Equation(s):$\displaystyle \left(\frac{\partial^2 w}{\partial x\partial y}\right)^2-
\frac{\partial^2 w}{\partial x^2}\frac{\partial^2 w}{\partial y^2}+
\left[f(x)\frac{\partial w}{\partial x}+f'_x(x)w\right]\frac{\partial^2 w}{\partial y^2}-
f^2(x)\left(\frac{\partial w}{\partial y}\right)^2=0$.\hfill\break
Solution(s),
Transformation(s),
Integral(s)
:
The transformation
$$
U=\frac{\partial w}{\partial y}+\int f(x)\,dx,
\quad V=\frac{\partial w}{\partial x}-f(x)w, \quad
T=S\left(x,y,w,\frac{\partial w}{\partial x},\frac{\partial w}{\partial y}\right),
$$
leads to the equation
$$
\frac{\partial U}{\partial T}=0.
$$
Its solution is $\,\displaystyle{U(V,T)=\Psi(V)}$.
Hence, we have a first order PDE:
$$
\frac{\partial w}{\partial y}+\int f(x)\,dx
=\Phi\left(\frac{\partial w}{\partial x}-f(x)w\right),
$$
where $\,\Phi\,$ is an arbitrary function.
Remarks:This result was obtained in co-authorship with L.V. Linchuk,
using generalized group analysis. The main idea of the method 
is outlined at EqWorld (http://eqworld.ipmnet.ru\slash 
ru/solutions/interesting.htm, in Russian).
Novelty:New equation(s) & solution(s) & transformation(s)
Author/Contributor's Details
Last name:Zaitsev
First name:Valentin
Middle(s) name:Feodorovich
Country:Russia
City:Saint-Petersburg
Statistic information
Submission date:Wed 09 Jan 2008 13:29
Edits by author:0

Edit (Only for author/contributor)


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