Questions: Considere la función u(x, y) = (x^2 y^2) / (x+y). Entonces el valor de la constante a para el cual se satisface la igualdad x (∂u/∂x) + y (∂u/∂y) = a u, es:

Solution Steps

To solve for the constant \( a \) in the given equation, we need to follow these steps:

1. Compute the partial derivatives of \( u(x, y) \) with respect to \( x \) and \( y \).
2. Substitute these partial derivatives into the equation \( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = a u \).
3. Simplify the equation to solve for the constant \( a \).

La función dada es \( u(x, y) = \frac{x^2 y^2}{x + y} \). Calculamos las derivadas parciales de \( u \) con respecto a \( x \) y \( y \):

\[ \frac{\partial u}{\partial x} = -\frac{x^2 y^2}{(x + y)^2} + \frac{2xy^2}{x + y} \]

\[ \frac{\partial u}{\partial y} = -\frac{x^2 y^2}{(x + y)^2} + \frac{2x^2 y}{x + y} \]

Sustituimos las derivadas parciales en la ecuación \( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = a u \):

\[ x \left(-\frac{x^2 y^2}{(x + y)^2} + \frac{2xy^2}{x + y}\right) + y \left(-\frac{x^2 y^2}{(x + y)^2} + \frac{2x^2 y}{x + y}\right) = a \frac{x^2 y^2}{x + y} \]

Al simplificar la ecuación, encontramos que el valor de \( a \) es:

\[ a = 3 \]

Final Answer