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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:4.2. Second-Order Quasilinear Hyperbolic Equations
Equation(s):\noindent
$\displaystyle \left(w\frac{\partial^2 w}{\partial x  \partial
y}+\frac{\partial w}{\partial x}\frac{\partial w}{\partial
y}\right)^2=\left[w\frac{\partial^2 w}{\partial
x^2}+\left(\frac{\partial w}{\partial
x}\right)^2-1\right]\left[w\frac{\partial^2 w}{\partial
y^2}+\left(\frac{\partial w}{\partial y}\right)^2-1\right]\,$.
Solution(s),
Transformation(s),
Integral(s)
:
\noindent
The general solution:\hfill\break
$\displaystyle w(x,y)=\pm \sqrt{2xG(W)+2F(W)-2yW+x^2+y^2}\,$,\hfill\break

where $W=W(t,x)$ is any solution of the following transcendental
equation
\hfill\break$\displaystyle xG\,'(W)+F\,'(W)-y=0$\hfill\break
and $F(z)$ and $G(z)$ are arbitrary functions.
Remarks:It is the Monge-Ampere equation, which defines the congruence of cycles of
constant curvature $K(x, y) = 1$ (see References).
Novelty:New solution(s) / integral(s)
References:V. Dryuma, On geometry of gonometric family of cycles. arXiv:0710.1581v1 [nlin.SI], (2007).
Author/Contributor's Details
Last name:Kosovtsov
First name:Yurii
Country:Ukraine
City:Lvov
Statistic information
Submission date:Wed 13 Feb 2008 11:01
Edits by author:0

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