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Equation data
Category:2. First-Order Partial Differential Equations
Subcategory:2.1. Linear Equations
Equation(s):$\displaystyle \frac{\partial w}{\partial t}+y\frac{\partial
w}{\partial x}+ f(t)\frac{\partial w}{\partial y}=0$.\hfill\break
Solution(s),
Transformation(s),
Integral(s)
:
\noindent $1^\circ$. Principal integrals: \hfill\break
$\displaystyle u_1=x-ty+t\int f(t)\,dt-\iint f(t)\,dt\,dt$, \\
$\displaystyle u_2=y-\int f(t)dt$.\\

$2^\circ$. General solution: $w=\Phi(u_1,u_2)$, where $\Phi$ is an
arbitrary function. \\

$3^\circ$. Solution with the initial condition
$w(t_0,x,y)=\Phi(x,y)$:
 $$
 w(t,x,y)=\Phi\left(x-y(t-t_0)+\int^t_{t_0}\int^t_{t'}f(t'')\,dt''dt',\
 y-\int^t_{t_0} f(t')\,dt'\right).
 $$
Novelty:Material has been partially published elsewhere
References:This equation is a particular case of the equation 6.8.1.4 of
Zaitsev V.F. and Polyanin A.D., Handbook of First Order Partial Differential Equations, Fizmatlit, Moscow, 2003.
Author/Contributor's Details
Last name:Sergeev
First name:Igor
Country:Russia
City:Moscow
Affiliation:IZMIRAN
Statistic information
Submission date:Mon 08 Feb 2010 15:11
Edits by author:0

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