Question
Prove that, if $z=2 x y+x . f\left(\frac{y}{x}\right)$ then $x \frac{\partial z}{\partial x}+y \frac{\partial z}{\partial y}=z+2 x y$.
Step 1
Given $z=2 x y+x . f\left(\frac{y}{x}\right)$, we can write this as $z=2xy + xf(y/x)$. The partial derivative of $z$ with respect to $x$ is given by: \[\frac{\partial z}{\partial x} = 2y + f\left(\frac{y}{x}\right) - Show more…
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Key Concepts
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Prove that, if $z=2 x y+x f\left(\frac{y}{x}\right)$ then $x \frac{\partial z}{d x}+y \frac{\partial z}{\partial y}=z+2 x y$.
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Further problems
If $z=x \cdot f\left(\frac{y}{x}\right)+F\left(\frac{y}{x}\right)$, prove that: (a) $x \frac{\partial z}{\partial x}+y \frac{\partial z}{\partial y}=z-F\left(\frac{y}{x}\right)$ (b) $x^{2} \frac{\partial^{2} z}{\partial x^{2}}+2 x y \frac{\partial^{2} z}{\partial x \cdot \partial y}+y^{2} \frac{\partial^{2} z}{\partial y^{2}}=0$
Further problems F.15
If $z=x . f\left(\frac{y}{x}\right)+F\left(\frac{y}{x}\right)$, prove that: (a) $x \frac{\partial z}{\partial x}+y \frac{\partial z}{\partial y}=z-F\left(\frac{\ell}{x}\right)$ (b) $x^{2} \frac{\partial^{2} z}{\partial x^{2}}+2 x y \frac{\partial^{2} z}{\partial x \cdot \partial y}+y^{2} \frac{\partial^{2} z}{\partial y^{2}}=0$
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