Question

Find a one-parameter family of solutions of the following differential equation, indicating carefully the interval in which the solutions are valid:
$left parenthesis sinx plus cosx right parenthesis space dy space equals space left parenthesis cosx minus sinx right parenthesis space dx$

Solution

The given differential equation is (sin x + cos x) dy = (cos x – sin x) dx

therefore space space space space space space space dy space equals space fraction numerator cosx minus sinx over denominator sinx plus cosx end fraction dx

Integrating,

integral 1 space dy space equals space integral fraction numerator cosx plus sinx over denominator sinx plus cosx end fraction dx

therefore space space space space space space space space space space space space space space space space space space straight y space equals log open vertical bar sinx plus cosx close vertical bar plus straight c comma space space space straight x space element of space straight R space is space the space solution

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