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View Equation

The database contains 327 equations (8 equations are awaiting activation).

Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.4. Other Second-Order Equations
Equation(s):$\displaystyle \left(\frac{\partial w}{\partial x}\right)^2\frac{\partial^2 w}{\partial y^2}-
\left(\frac{\partial w}{\partial y}\right)^2\frac{\partial^2 w}{\partial x^2}+
\frac{\partial w}{\partial x}\frac{\partial w}{\partial y}\left[g(y)\frac{\partial w}{\partial x}-
f(x)\frac{\partial w}{\partial y}\right]=0$.\hfill\break
Solution(s),
Transformation(s),
Integral(s)
:
The transformation
$$
U=w, \quad
P=\frac{\partial w}{\partial x}\exp\left(\int g(y)\,dy\right), \quad
Q=\frac{\partial w}{\partial y}\exp\left(\int f(x)\,dx\right),
$$
leads to the equation
$$
P\frac{\partial U}{\partial P}+Q\frac{\partial U}{\partial Q}=0.
$$
Its solution is $\,\displaystyle{U(P,Q)=\Psi\left(\frac{Q}{P}\right)}$.
Hence, we have a first order PDE:
$$
\frac{\partial w}{\partial x}\frac{\partial w}{\partial y}\exp\left(\int f(x)\,dx+\int g(y)\,dy\right)=
\Phi(w),
$$
where $\Phi$ is an arbitrary function.
Remarks:This result was obtained in co-authorship with L.V. Linchuk,
using generalized group analysis. The main idea of the method 
is outlined at EqWorld (http://eqworld.ipmnet.ru\slash 
ru/solutions/interesting.htm, in Russian).
Novelty:New equation(s) & solution(s) & transformation(s)
Author/Contributor's Details
Last name:Zaitsev
First name:Valentin
Middle(s) name:Feodorovich
Country:Russia
City:Saint-Petersburg
Statistic information
Submission date:Wed 09 Jan 2008 13:44
Edits by author:0

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