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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.2. Second-Order Quasilinear Hyperbolic Equations
Equation(s):$\displaystyle \left[g(w)\frac{\partial w}{\partial y}+f(y)\right]\frac{\partial^2 w}{\partial x\partial y}
-\left[g(w)\frac{\partial w}{\partial x}-h(x)\right]\frac{\partial^2 w}{\partial y^2}=0$,\hfill\break
The equation is represented in equivalent form
\frac{\partial U}{\partial T}=0,
U=\frac{\partial w}{\partial y}, \quad V=\int h(x)\,dx-\int f(y)\,dy-\int g(w)\,dw, \quad T=x,
The solution of Eq.(1) is $\,U=\Phi(V)$. Hence, we have a first order PDE:
\frac{\partial w}{\partial y}=\Phi\left(\int h(x)\,dx-\int f(y)\,dy-\int g(w)\,dw\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
Middle(s) name:Feodorovich
Statistic information
Submission date:Mon 02 Jul 2007 08:39
Edits by author:0

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