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:4.2. Second-Order Quasilinear Hyperbolic Equations
Equation(s):$\displaystyle \left[yw\frac{\partial w}{\partial y}+\frac{w^2}{2}+f(y)\right]\frac{\partial^2 w}{\partial x\partial y}-
\left(yw-\frac{\partial w}{\partial x}\right)\frac{\partial^2 w}{\partial y^2}=0$,\hfill\break
The transformation
U=\frac{\partial w}{\partial y}, \quad V=x-\int f(y)\,dy-\frac12 yw^2, 
\quad T=2y-w^2,
leads to the equation
\frac{\partial U}{\partial T}=0.
Its solution is $\,U=\Phi(V)$. Hence, we have a first order PDE:
\frac{\partial w}{\partial y}=\Phi\left(x-\int f(y)\,dy-\frac12 
where $\,\Phi(z)\,$ is arbitrary function.
Remarks:This result was received by generalized group analysis. The main idea of the method
is announced in
Novelty:New equation(s) & solution(s) / integral(s)
Author/Contributor's Details
Last name:Zaitsev
First name:Valentin
Middle(s) name:Feodorovich
Statistic information
Submission date:Fri 13 Jul 2007 07:33
Edits by author:1
Last edit by author:Wed 22 Aug 2007 17:23

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