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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.2. Second-Order Quasilinear Hyperbolic Equations
Equation(s):$\displaystyle \left(y^2w\frac{\partial w}{\partial y}+
ax\right)\frac{\partial^2 w}{\partial x\partial y}-
y\left(yw\frac{\partial w}{\partial x}-a\right)\frac{\partial^2 w}{\partial y^2}=0$.
The transformation
U=\frac{\partial w}{\partial y}, \quad V=\frac{ax-y^2}{ay}, \quad T=2y-w^2,
leads to the equation
2a\frac{\partial U}{\partial T}-\frac{\partial U}{\partial V}=0.
Its solution is $\,U=\Phi(2aV+T)$. Hence, we have a first order PDE:
\frac{\partial w}{\partial y}=\Phi\left(\frac{2ax-yw^2}{y}\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 17:27
Edits by author:0

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