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View Equation

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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.2. Second-Order Quasilinear Hyperbolic Equations
Equation(s):$\displaystyle \frac{\partial^2 w}{\partial x^2}+
\left[\frac{\partial w}{\partial x}-f(x)y\right]\frac{\partial^2 w}{\partial x\partial y}-
f(x)y\frac{\partial w}{\partial x}\frac{\partial^2 w}{\partial y^2}=0$.
Solution(s),
Transformation(s),
Integral(s)
:
The transformation
$$
U=\frac{\partial w}{\partial x}\exp\left(\frac{\partial w}{\partial y}\right),
\quad V=w-y, \quad T=y\exp\left(\int f(x)\,dx\right),
$$
leads to the equation
$$
\frac{\partial U}{\partial V}=0.
$$
Its solution is $\,U=\Phi(T)$. Hence, we have a first order PDE:
$$
\frac{\partial w}{\partial x}=
\exp\left(-\frac{\partial w}{\partial y}\right)\Phi\left(y\exp\left(\int f(x)\,dx\right)\right),
$$
where $\,\Phi(z)\,$ is arbitrary function.
Remarks:This result was received by generalized group analysis. The main idea of the method
is announced in http://eqworld.ipmnet.ru/ru/solutions/interesting.htm
Novelty:New equation(s) & transformation(s)
Author/Contributor's Details
Last name:Zaitsev
First name:Valentin
Country:Russia
City:Saint-Petersburg
Statistic information
Submission date:Tue 04 Sep 2007 17:47
Edits by author:0

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