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Ecuaciones diferenciales en derivadas parciales

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Problemas resueltos

Ecuaciones en derivadas parciales

Pasar a las nuevas variables independientes u, v y a la nueva función w la expresión
    \( \displaystyle \frac{\partial^2 z}{\partial x^2} + 2\frac{\partial^2 z}{\partial x \partial y} + \left(1 + \frac{x}{y}\right)\frac{\partial^2 z}{\partial y^2} =0 \)
donde:
    \( u = x \;;\; v = x+y \;;\; w = x+y+z\)
- Respuesta 75

Derivando z respecto de x e y pero considerando las variables intermedias u y v:
    \( \displaystyle \begin{array}{l}
    \frac{\partial w}{\partial x} = \frac{\partial w}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial w}{\partial v}\frac{\partial v}{\partial x} = \frac{\partial w}{\partial v} +\frac{\partial w}{\partial u}\\
     \\
    \frac{\partial w}{\partial y} = \frac{\partial w}{\partial u}\frac{\partial u}{\partial y} + \frac{\partial w}{\partial v}\frac{\partial v}{\partial y} = \frac{\partial w}{\partial v}
    \end{array} \)
Por otro lado, considerando a z como variable intermedia, la diferencial de w vale:
    \( \displaystyle dw = \frac{\partial w}{\partial x}dx + \frac{\partial w}{\partial y}dy = \left(1 + \frac{\partial z}{\partial x}\right)dx + \left(1 + \frac{\partial z}{\partial y}\right)dy \)
Y, por tanto:
    \( \displaystyle \frac{\partial w}{\partial x} = 1 +\frac{\partial z}{\partial x}\quad ; \quad \frac{\partial w}{\partial y} = 1+ \frac{\partial z}{\partial y} \)
Si igualamos estas expresiones con las anteriores resulta:
    \( \displaystyle \begin{array}{l}
    \frac{\partial w}{\partial x} = \frac{\partial w}{\partial v} +\frac{\partial w}{\partial u}= 1 +\frac{\partial
    z}{\partial x} \Rightarrow \frac{\partial z}{\partial x}= \frac{\partial w}{\partial v} +\frac{\partial w}{\partial u}- 1 \\
     \\
    \frac{\partial w}{\partial y} = \frac{\partial w}{\partial v}= 1 +\frac{\partial z}{\partial y}\Rightarrow \frac{\partial z}{\partial y} = \frac{\partial w}{\partial v} - 1
    \end{array} \)
Derivamos ahora estas expresiones con los que nos quedara:
    \( \displaystyle \begin{array}{l}
    \frac{\partial }{\partial x}\left( \frac{\partial z}{\partial x} \right) = \frac{\partial }{\partial x}\left( \frac{\partial w}{\partial v} +\frac{\partial w}{\partial u}- 1 \right) = \frac{\partial^2 w}{\partial u^2}\frac{\partial u}{\partial x} + \frac{\partial^2 u}{\partial v\partial u}\frac{\partial w}{\partial x} + \\
     \\
    + \frac{\partial^2 w}{\partial u \partial v}\frac{\partial u}{\partial x} + \frac{\partial^2 w}{\partial v^2}\frac{\partial v}{\partial x}= \frac{\partial^2 w}{\partial u^2} + 2\frac{\partial^2 w}{\partial u \partial v}+\frac{\partial^2 w}{\partial v^2} \\
     \\
    \frac{\partial }{\partial y}\left( \frac{\partial z}{\partial y} \right) = \frac{\partial }{\partial y}\left( \frac{\partial w}{\partial v} - 1 \right) = \frac{\partial^2 w}{\partial u \partial v}\frac{\partial u}{\partial y}+\frac{\partial^2 w}{\partial v^2}\frac{\partial v}{\partial y}=\frac{\partial^2 w}{\partial v^2} \\ \\
    \frac{\partial }{\partial x}\left( \frac{\partial z}{\partial y} \right) = \frac{\partial }{\partial x}\left( \frac{\partial w}{\partial v} - 1 \right) = \frac{\partial^2 w}{\partial u \partial v}\frac{\partial u}{\partial x}+\frac{\partial^2 w}{\partial v^2}\frac{\partial v}{\partial x}= \\ \\= \frac{\partial^2 w}{\partial u \partial v}+\frac{\partial^2 w}{\partial v^2}
    \end{array} \)
Multiplicando cada expresión por los términos indicados resulta:
    \( \displaystyle \begin{array}{l}
    \frac{\partial^2 w}{\partial u^2} + \frac{\partial^2 w}{\partial u \partial v}+ \frac{\partial^2 w}{\partial v^2}- 2\left( \frac{\partial^2 w}{\partial u \partial v}+ \frac{\partial^2 w}{\partial v^2} \right)+ \\
     \\
    + \left(1 + \frac{x}{y}\right)\frac{\partial^2 w}{\partial v^2}= \frac{\partial^2 w}{\partial u^2} +\left(\frac{x}{y}\right)\frac{\partial^2 w}{\partial v^2}=0
    \end{array} \)
Y considerando la relación:
    \( \displaystyle u=x\quad;\quad v-u = y\Rightarrow \frac{x}{y} = \frac{u}{v-u} \)
Resulta finalmente:
    \( \displaystyle \frac{\partial^2 w}{\partial u^2} + \frac{u}{v-u} \frac{\partial^2 w}{\partial v^2}=0 \)
EJERCICIOS-ECUACIONES EN DERIVADAS PARCIALES
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