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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:4.2. Second-Order Quasilinear Hyperbolic Equations
Equation(s):$\displaystyle \left[f(y)+\frac{\partial w}{\partial y}\right]\frac
{\partial^2 w}{\partial x\partial y}-
\left[g(x)+\frac{\partial w}{\partial x}\right]\frac
{\partial^2 w}{\partial^2 y}=0$.
Solution(s),
Transformation(s),
Integral(s)
:
The equation is represented in equivalent form
$$ 
\frac{\partial U}{\partial y}\frac{\partial V}{\partial x}-
\frac{\partial U}{\partial x}\frac{\partial V}{\partial y}=0,
\eqno(1)
$$
where 
$$
U=\int f(y)\,dy+\int g(x)\,dx+w,\quad V=\frac{\partial w}{\partial y}.
\eqno(2)
$$
The fact that the Jacobian of the two functions $U$ and $V$ is zero 
(see Eq.(1)) means that the two functions (2) are functionally
dependent. Hence, $V$ must be a function of~$U$, so that
we have a first order PDE:
$$
\frac{\partial w}{\partial y}=
\Phi\left(\int f(y)\,dy+\int g(x)\,dx+w\right),
$$
where $\Phi(z)$ is an arbitrary function.
Remarks:The equation is represented in conservation law form:
$$
\frac{\partial V}{\partial w}-\frac{\partial V}{\partial T}=0,
\eqno(3)
$$
where
$$
V=\frac{\partial w}{\partial y}, \quad T=\int f(y)\,dy+\int
g(x)\,dx,
$$
The solution of Eq.(3) is $\,V=\Phi(w+T)$. Hence, we have a first order
PDE:
$$
\frac{\partial w}{\partial y}=\Phi\left(\int f(y)\,dy+\int 
g(x)\,dx+w\right),
$$
where $\,\Phi(z)\,$ is arbitrary function.
Novelty:New equation(s) & solution(s) / integral(s)
Author/Contributor's Details
Last name:Zaitsev
First name:Valentin
Middle(s) name:Feodorovich
Country:Russia
Statistic information
Submission date:Sat 23 Jun 2007 14:32
Edits by author:0

Edit (Only for author/contributor)


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