Differential Equations

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Question

Find a one parameter family of solutions of each of the following differential equation e^–y sec² y dy = dx + x dy.

Solution

The given differential equation is
e^–y sec² y dy = dx + x dy

therefore space space space left parenthesis straight e to the power of negative straight y end exponent space sec squared straight y minus straight x right parenthesis space dy space equals space dx space space space space space space space space space space space or space space space space dx over dy space equals space straight e to the power of negative straight y end exponent space sec squared straight y minus straight x therefore space space space space space space space space dy over dx plus straight x space equals space straight e to the power of negative straight y end exponent space secy

Comparing it with

dx over dy plus Px space equals space straight Q comma space space we space get space straight P space equals space 1 comma space space straight Q space equals space straight e to the power of negative straight y end exponent space sec squared straight y

integral straight P space dy space equals space integral 1 space dy space equals space straight y. space straight e to the power of integral straight P space dy end exponent space equals space straight e to the power of straight y

therefore

solution of differential equation is

xe to the power of integral straight P space dy end exponent space equals space integral Qe to the power of integral straight P space dy end exponent plus straight c space space space space space space or space space space space space straight x space straight e to the power of straight y space equals space integral straight e to the power of negative straight y end exponent space sec squared straight y. space straight e to the power of straight y space dy space plus straight c

straight x space straight e to the power of straight y space equals space integral sec squared straight y space dy space plus space straight c space space space space space space or space space space straight x space straight e to the power of straight y space equals space tan space straight y space plus straight c