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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.2. Second-Order Quasilinear Hyperbolic Equations
Equation(s):$\displaystyle \left(\frac{\partial w}{\partial y}+
ae^{x}\right)\frac{\partial^2 w}{\partial x\partial y}+
\left(w-\frac{\partial w}{\partial x}\right)\frac{\partial^2 w}{\partial y^2}=0$.
The transformation
U=\frac{\partial w}{\partial y}, \quad V=e^{ay}(x-w), \quad T=w^2-2y,
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(y+\frac1a e^{-x}w\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
Novelty:New equation(s) & solution(s) / integral(s)
Author/Contributor's Details
Last name:Zaitsev
First name:Valentin
Statistic information
Submission date:Wed 22 Aug 2007 17:32
Edits by author:0

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