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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):\noindent
$\displaystyle \frac{\partial^2 w}{\partial x\partial y}=f\left(\frac{\partial w}{\partial x}\right)\frac{\partial w}{\partial y}$.
Solution(s),
Transformation(s),
Integral(s)
:
\noindent
1. Suppose $w(x,y)$ is a solution of the equation in question. Then the function
$$
w_1=w(x+C,\,\varphi(y)),
$$
where $C$ is an arbitrary constant and $\varphi(y)$ is an arbitrary function,
is also a solution of the equation.
\medskip

\noindent
2. Exact solution:
$$
w=U(z),\quad z=x+\varphi(y),
$$
where $\varphi(y)$ is an arbitrary function, 
and the function $U(z)$ is defined parametrically by
$$
z=\int\frac {d\xi}{\xi f(\xi)}+ C_1,\quad \ U=\int\frac {d\xi}{f(\xi)}+ C_2,
$$ 
$C_1$ and $C_2$ are arbitrary constants.
\medskip

\noindent
3. First integral:
$$
\int^{w_x}_a\frac{du}{f(u)}=w+\theta(x), 
$$ 
where $w_x$ is the partial derivative of $w$ with respect to $x$,
$\theta(x)$ is an arbitrary function, and $a$ is an arbitrary constant.
The first integral may be treated as a first-order ordinary differential
equation in $x$. On finding its general solution, one should replace the
constant of integration $C$ with an arbitrary function of time
$\psi(t)$, since $w$ is dependent on $x$ and~$t$.
Novelty:New solution(s) / integral(s)
Author/Contributor's Details
Last name:Polyanin
First name:Andrei
Country:Russia
Statistic information
Submission date:Thu 14 Dec 2006 14:35
Edits by author:0

Edit (Only for author/contributor)


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