Solve:
tan y dx + sec² y tan x dy = 0.

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Solution

The given differential equation is
tan y dx + sec² y tan x dy = 0 or sec² y tan x dy = - tan y dx
Separating the variables, we get,

fraction numerator sec squared straight y over denominator tan space straight y end fraction dy space equals space minus fraction numerator 1 over denominator tan space straight x end fraction dx

Integrating,

integral fraction numerator sec squared straight y over denominator tan space straight y end fraction dy space equals negative fraction numerator 1 over denominator tan space straight x end fraction dx

therefore space space log space open vertical bar tan space straight y close vertical bar space equals space minus log space open vertical bar sin space straight x close vertical bar space plus space log space straight c apostrophe therefore space space log space open vertical bar tan space straight y close vertical bar space plus space log space open vertical bar sin space straight x close vertical bar space equals space log space straight c apostrophe therefore space space log space open vertical bar tan space straight y space sinx space close vertical bar space equals space log space straight c apostrophe therefore space space open vertical bar tany space sinx close vertical bar space equals space straight c apostrophe therefore space space tany space sinx space space equals space straight A comma space where space straight A space is space constant. space

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