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Equation data
Category:3. Linear Partial Differential Equations
Subcategory:3.4. Other Second-Order Equations
Equation(s):$\displaystyle \frac{\partial^2 w}{\partial x^2} + 
a(x)\frac{\partial^2 w}{\partial y^2} = 0$,\hfill\break
where $a(x)$ in an integrable function.\\
Previous PDE is a generalized Tricomi equation.
Solution(s),
Transformation(s),
Integral(s)
:
Solution:\hfill\break

$\displaystyle w(x,y) = \int_b^y t(x,r) dr - \int_a^x \int_a^q \left[a(s) \hspace{0.1cm} t_y(s,b) \right] ds \hspace{0.1cm} dq$,\hfill\break

where $a$, $b$ are any real numbers, $t(x,y)$ is any other solution of previous PDE itself; this formula 
can be used as a {\it solutions machine}: starting from the most simple 
not null solution, that is from $t(x,y) = 1$, it can be used for 
iterative construction of solutions.\\

{\it Example}. Let be $w_{xx} + \textnormal{cos}(x) w_{yy} = 0$ a generalized Tricomi equation. Note that $t(x,y) = y$ is a (trivial) solution. 
Then, from previous formula with $a=b=0$, the following function is a solution:

\begin{equation}
	w(x,y) = \int_0^y r \hspace{0.1cm} dr - \int_0^x \int_0^q \textnormal{cos}(s) \hspace{0.1cm} ds \hspace{0.1cm} dq = \frac{1}{2}y^2 -1 + \textnormal{cos}(x)
\end{equation}
Remarks:Full paper with proof:\\

G.Argentini, {\it Construction of classic exact solutions for Tricomi equation},
arXiv, math.AP, 1010.1681, 8/10/2010.\\
Novelty:Material has been fully published elsewhere
References:arXiv, 2010.
Author/Contributor's Details
Last name:Argentini
First name:Gianluca
Country:Italy
Affiliation:Riello Burners
Statistic information
Submission date:Mon 11 Oct 2010 10:36
Edits by author:0

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