Question

Find a one parameter family of solutions of each of the following differential equation:
y' = (cos²x – sin²x) cos² y

Solution

The given differential equation is

dy over dx space equals space left parenthesis cos squared straight x minus sin squared straight x right parenthesis space cos squared straight y

Separating the variables, we get,

fraction numerator 1 over denominator cos squared straight y end fraction dy space equals space left parenthesis cos squared straight x minus sin squared straight x right parenthesis space dx

Integrating,

integral sec squared straight y space dy space equals space integral cos space 2 straight x space dx

therefore space space space space space space space space space tan space straight y space equals space fraction numerator sin space 2 straight x over denominator 2 end fraction plus straight c

is the required solution.

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