Question

If w= f{ xy/(x^2 + y^2) } show that x.∂w/∂x + y.∂w/∂y = 0

Answer 1

Answer on Question #49497 – Math – Differential Calculus | Equations

Task:

If $w = \{xy / (x^2 + y^2) \}$ show that $x \frac{\partial w}{\partial x} + y \frac{\partial w}{\partial y} = 0$.

Solution:

$$\frac{\partial w}{\partial x} = \frac{y(x^2 + y^2) - 2x^2y}{(x^2 + y^2)^2}$$ $$\frac{\partial w}{\partial y} = \frac{x(x^2 + y^2) - 2y^2x}{(x^2 + y^2)^2}$$ $$\begin{aligned} x \frac{\partial w}{\partial x} + y \frac{\partial w}{\partial y} &= \frac{xy(x^2 + y^2) - 2x^3y}{(x^2 + y^2)^2} + \frac{yx(x^2 + y^2) - 2y^3x}{(x^2 + y^2)^2} = \frac{2yx(x^2 + y^2) - 2x^3y - 2y^3x}{(x^2 + y^2)^2} = \\ &= 2xy \frac{(x^2 + y^2) - x^2 - y^2}{(x^2 + y^2)^2} = 0 \end{aligned}$$

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