Question
Let $z=x^{c} e^{-y / x}$, where $c$ is a constant. Find the value of $c$ such that$$\frac{\partial z}{\partial x}=y \frac{\partial^{2} z}{\partial y^{2}}+\frac{\partial z}{\partial y}$$
Step 1
The partial derivative of $z$ with respect to $x$ is given by: \[ \frac{\partial z}{\partial x} = c x^{c-1} e^{-y/x} - y x^{c-2} e^{-y/x} \] The partial derivative of $z$ with respect to $y$ is given by: \[ \frac{\partial z}{\partial y} = -x^{c} e^{-y/x} \cdot Show more…
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