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Differential Equations

Question

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Find one-parameter families of solution curves of the following differential equation:
$straight y apostrophe space minus space straight y space equals space cos space straight x$

Solution

The given differential equation is

straight y apostrophe minus straight y space equals space cos space straight x space space space space space space or space space space space space dy over dx minus straight y space equals space cos space straight x

Comparing it with

dy over dx plus Py space equals space straight Q comma space space we space space get space comma space space straight P space equals space minus 1 comma space space space straight Q space equals space cos space straight x space

integral space thin space straight P space dx space equals space minus integral 1 space dx space equals space minus straight x comma space space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of negative straight x end exponent

solution is

space straight y space straight e to the power of integral straight P space dx end exponent space equals space integral straight Q space straight e to the power of integral space straight P space dx end exponent space dx plus straight c

or

straight y space straight e to the power of negative straight x end exponent space equals space integral space left parenthesis cosx right parenthesis. space straight e to the power of negative straight x end exponent dx plus straight c

therefore space space space space space space space space space space straight y space straight e to the power of negative straight x end exponent space equals space fraction numerator 1 over denominator 1 plus 1 end fraction straight e to the power of negative straight x end exponent space left parenthesis sin space straight x minus space cos space straight x right parenthesis space plus space straight c therefore space space space space space space space space space space space space space space straight y space space equals 1 half left parenthesis sin space straight x space minus space cos space straight x right parenthesis space plus space ce to the power of straight x

which is required solution.

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