Differential Equations

Question

Solve:
$straight y apostrophe plus straight y space equals space sin space straight x$

Solution

The given differential equation is $dy over dx plus straight y space equals space sin space straight x$
Comparing it with $dy over dx plus straight P space straight y space space equals straight Q comma space space we space get space comma space straight P space equals space 1 comma space straight Q equals space sin space straight x$
$therefore space space space space space space space integral straight P space dx space equals space integral 1 space dx space equals space straight x comma space space space space straight I. straight F. space equals space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of straight x$
$therefore space$ solution of differential equation is
$straight y. space straight e to the power of straight x space equals integral sinx. space straight e to the power of straight x space dx plus straight c space space space or space space space straight y. space straight e to the power of straight x space equals space fraction numerator 1 over denominator square root of 1 plus 1 end root end fraction straight e to the power of straight x sin space open parentheses straight x minus tan to the power of negative 1 end exponent 1 over 1 close parentheses plus straight c space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space integral straight e to the power of ax space sin space bx space equals space minus fraction numerator 1 over denominator square root of straight a squared plus straight b squared end root end fraction straight e to the power of ax space sin space open parentheses bx minus tan to the power of negative 1 end exponent straight b over straight a close parentheses close square brackets or space space space space straight y space equals fraction numerator 1 over denominator square root of 2 end fraction sin space open parentheses straight x minus straight pi over 4 close parentheses plus space ce to the power of negative straight x end exponent space space space space space space space$

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Differential Equations

Solve:
$straight y apostrophe plus straight y space equals space sin space straight x$

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