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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):$\displaystyle \left[yw\frac{\partial w}{\partial y}+\frac{w^2}{2}+f(y)\right]\frac{\partial^2 w}{\partial x\partial y}-
\left(yw-\frac{\partial w}{\partial x}\right)\frac{\partial^2 w}{\partial y^2}=0$,\hfill\break
Solution(s),
Transformation(s),
Integral(s)
:
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:
$$
\frac{\partial w}{\partial y}=\Phi\left(x-\int f(y)\,dy-\frac12 
yw^2\right),
$$
where $\,\Phi(z)\,$ is arbitrary function.
Remarks:This result was received by generalized group analysis. The main idea of the method
is announced in http://eqworld.ipmnet.ru/ru/solutions/interesting.htm
Novelty:New equation(s) & solution(s) / integral(s)
Author/Contributor's Details
Last name:Zaitsev
First name:Valentin
Middle(s) name:Feodorovich
Country:Russia
City:Saint-Petersburg
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)


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