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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.2. Second-Order Quasilinear Hyperbolic Equations
Equation(s):$\displaystyle \frac{\partial w}{\partial y}\frac{\partial^2 w}{\partial x^2}-
\frac{\partial w}{\partial x}\frac{\partial^2 w}{\partial x\partial y}-
f(y)\frac{\partial w}{\partial x}=0$.
The transformation
U=\exp\left[\frac{\partial w}{\partial x}+\int f(y)\,dy\right], \quad V=w, \quad T=ax+by,
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 x}=\Phi(w)-\int f(y)\,dy,
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 17:38
Edits by author:0

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