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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):$\displaystyle \left(\frac{\partial^2 w}{\partial x\partial y}\right)^2-
\frac{\partial^2 w}{\partial x^2}\frac{\partial^2 w}{\partial y^2}-
\left[f\left(x,\frac{\partial w}{\partial y}\right)\right]^2=0$,\hfill\break
where $f(x,z)$ is represented in implicit form
$\displaystyle \Psi\left(xf+\frac{\partial w}{\partial y},f\right)=0$, $\Psi$ is arbitrary
function.
Solution(s),
Transformation(s),
Integral(s)
:
The transformation
$$
U=\frac{1}{f\left(x,\frac{\partial w}{\partial y}\right)}, \quad V=y, \quad T=\frac{\partial w}{\partial x},
$$
leads to the equation
$$
\frac{\partial U}{\partial T}=U\frac{\partial U}{\partial V},
$$
Its solution is $\,\displaystyle{T=\frac{V}{U}+\Phi\left(\frac{1}{U}\right)}$.
Hence, we have a first order PDE:
$$
f\left(x,\frac{\partial w}{\partial y}\right)y+
\Phi\left(f\left(x,\frac{\partial w}{\partial y}\right)\right)=\frac{\partial w}{\partial x},
$$
where $\,\Phi(z)\,$ is arbitrary function.

The special case $f=a=\mathrm{const}$: we have a first order PDE
$$
\Omega\left(\frac{\partial w}{\partial x}-ay,\,\, \frac{\partial w}{\partial y}+ax\right)=0,
$$
where $\,\Omega(z_1,z_2)\,$ 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) & solution(s) / integral(s)
Author/Contributor's Details
Last name:Zaitsev
First name:Valentin
Country:Russia
City:Saint-Petersburg
Statistic information
Submission date:Tue 04 Sep 2007 17:58
Edits by author:0

Edit (Only for author/contributor)


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