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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.2. Second-Order Quasilinear Hyperbolic Equations
Equation(s):$\displaystyle \frac{\partial w}{\partial y}\frac{\partial^2 w}{\partial x^2}-
\frac{\partial w}{\partial x}\frac{\partial^2 w}{\partial x\partial y}-
[f(x)+g(y)]\left(\frac{\partial w}{\partial x}\right)^2+
f'(x)y\frac{\partial w}{\partial x}\frac{\partial w}{\partial y}=0$.
The transformation
U=\exp\left[w+\int g(y)\,dy\right]\frac{\partial w}{\partial x}, \quad V=w, \quad T=f(x)y,
leads to the equation
\frac{\partial U}{\partial T}+U=0.
Its solution is $\,U=\Phi(V)e^{-T}$. Hence, we have a first order PDE:
\frac{\partial w}{\partial x}=\exp\left[-f(x)y-\int g(y)\,dy\right]\Phi(w),
where $\,\Phi(w)\,$ 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 17:41
Edits by author:0

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