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Differential Equations

Question

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For the differential equation, find the general solution:
$dy over dx plus 2 straight y space equals sin space straight x$

Solution

The given different equation is

dy over dx plus 2 straight y space equals space sin space straight x

Comparing it with

dy over dx plus straight P space straight y space space equals space straight Q comma space we space get comma space

P = 2, Q = sin x

rightwards double arrow space

integral straight P space dx space equals space integral space 2 space dx space equals space 2 straight x

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 2 straight x end exponent

therefore

solution of differential equation is

straight y. space straight e to the power of 2 straight x end exponent space equals space integral space sinx. space straight e to the power of 2 straight x end exponent space dx plus straight c space space space space space space space space space space open square brackets because space space space straight y space straight e to the power of integral Pdx end exponent space equals space integral straight Q space straight e to the power of integral straight P space dx end exponent dx plus straight c close square brackets

or

straight y. space straight e to the power of 2 straight x end exponent space equals space fraction numerator 1 over denominator square root of 4 plus 1 end root end fraction straight e to the power of 2 straight x end exponent space sin space open parentheses straight x minus tan to the power of negative 1 end exponent 1 half close parentheses plus straight c

open square brackets because space space integral space space straight e space to the power of ax space sin space bx space dx space equals space fraction numerator 1 over denominator square root of straight a squared plus straight b squared end root end fraction space 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

straight y equals space fraction numerator 1 over denominator square root of 5 end fraction sin space open parentheses straight x minus tan to the power of negative 1 end exponent 1 half close parentheses plus ce to the power of negative 2 straight x end exponent

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