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Find an implicit formula satisfied by solutions of (tan x)(cos y) = -y′ tan y

Assume solutions of the equation

    \[ \tan x \cos y = -y' \tan y. \]

exist and find an implicit formula satisfied by these solutions.


This is a separable first order equation. We compute

    \begin{align*}  \tan x \cos y = -y' \tan y && \implies && \tan x &= -y' \frac{\sin y}{\cos^2 y} \\  && \implies && \int \tan x \, dx &= -\int \frac{\sin y}{\cos^2 y} \, dy \\  && \implies && -\log |\cos x| + C &= \frac{1}{\cos y} \\  && \implies && \cos x &= C e^{\frac{1}{\cos y}}.  \end{align*}

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