The World of Mathematical Equations

IPM Logo

Exact Solutions Methods Software For Authors Math Forums

EqArchive: Add Equation/Solution > View Equation

 English only

View Equation

The database contains 327 equations (9 equations are awaiting activation).

Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.4. Other Second-Order Equations
Equation(s):$\displaystyle \left(\frac{\partial^2 w}{\partial x\partial y}\right)^2-
\frac{\partial^2 w}{\partial x^2}\frac{\partial^2 w}{\partial y^2}-
2\frac{\partial^2 w}{\partial x^2}+(a+1)\frac{\partial^2 w}{\partial x\partial y}+a=0$.\hfill\break
The transformation
U=\frac{\partial w}{\partial y}+x+2y,
\quad V=\frac{\partial w}{\partial x}+y, \quad T=\frac{\partial w}{\partial y}+x,
leads to the equation
\frac{\partial U}{\partial T}-\frac{1-a}{2}\frac{\partial U}{\partial V}=0.
Its solution is $\,\displaystyle{U(V,T)=\Psi\left(T+\frac{2}{a-1}V\right)}$.
Hence, we have a first order PDE:
\frac{\partial w}{\partial y}+x+2y=
\Phi\left((1-a)\left(\frac{\partial w}{\partial y}+x\right)+2\frac{\partial w}{\partial x}+
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 (\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
Statistic information
Submission date:Wed 09 Jan 2008 13:14
Edits by author:0

Edit (Only for author/contributor)

The EqWorld website presents extensive information on solutions to various classes of ordinary differential equations, partial differential equations, integral equations, functional equations, and other mathematical equations.

Copyright © 2006-2011 Andrei D. Polyanin, Alexei I. Zhurov and Alexander L. Levitin