If $u = \frac{y}{x} + \frac{z}{x} + \frac{x}{y}$ $u= y/x+ z/x + x/y$ , show that $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z} = 0$ $x(partial u)/(partial x)+y(partial u)/(partial y)+z(partial u)/(partial z)=0$ ?

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If $u = \frac{y}{x} + \frac{z}{x} + \frac{x}{y}$ $u= y/x+ z/x + x/y$ , show that $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z} = 0$ $x(partial u)/(partial x)+y(partial u)/(partial y)+z(partial u)/(partial z)=0$ ?

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Answer 1 · 2018-04-05T17:40:26

We have:

$u = \frac{y}{x} + \frac{z}{x} + \frac{x}{y}$ $u = y/x+z/x+x/y$

and we seek to validate that $f$ $f$ satisfies the Partial differential Equation:

$x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z}$ $x(partial u)/(partial x) + y (partial u)/(partial y) + z (partial u)/(partial z)$

(In other words we are validating that a solution to the given PDE is $u$ $u$ ). We compute the partial derivative (by differentiating wrt to specified variable and treating all other variables as constants), and applying the chain rule:

$u_{x} = \frac{\partial u}{\partial x} = - \frac{y}{x^{2}} - \frac{z}{x^{2}} + \frac{1}{y}$ $u_x = (partial u)/(partial x) =-y/x^2-z/x^2 + 1/y$

$u_{y} = \frac{\partial u}{\partial y} = \frac{1}{x} - \frac{x}{y^{2}}$ $u_y = (partial u)/(partial y) = 1/x-x/y^2$

$u_{z} = \frac{\partial u}{\partial z} = \frac{1}{x}$ $u_z = (partial u)/(partial z) = 1/x$

Next we compute the LHS of the desired expression:

$L H S = x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z}$ $LHS = x(partial u)/(partial x) + y (partial u)/(partial y) + z (partial u)/(partial z)$

$\ \ \ \ \ \ \ \ = x((y)/(1-2xy+y^2)^(2)) - y ((x-y)/(1-2xy+y^2)^(2))$

$\ \ \ \ \ \ \ \ = x(-y/x^2-z/x^2 + 1/y) + y(1/x-x/y^2) + z(1/x)$

$\ \ \ \ \ \ \ \ = -y/x-z/x + x/y + y/x-x/y + z/x$

Noting that all terms cancel, we then have the desired result:

$LHS = 0$
$\ \ \ \ \ \ \ \ = RHS \ \ \$ QED

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If $u = \frac{y}{x} + \frac{z}{x} + \frac{x}{y}$ $u= y/x+ z/x + x/y$ , show that $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z} = 0$ $x(partial u)/(partial x)+y(partial u)/(partial y)+z(partial u)/(partial z)=0$ ?

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