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Differential Equations

Question

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Solve:
$dy over dx plus 2 straight y space equals space sin space 5 straight x$

Solution

The given differential equation is

dy over dx plus 2 straight y space equals space sin space 5 straight x

Comparing it with

dy over dx plus Py space equals space straight Q comma space we space get comma

P = 2, Q = sin 5x

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

or

straight y space straight e to the power of 2 straight x end exponent space equals space integral space sin space 5 straight x. space straight e to the power of 2 straight x end exponent plus straight c

or

ye to the power of 2 straight x end exponent space equals space fraction numerator 1 over denominator square root of 4 plus 25 end root end fraction straight e to the power of 2 straight x end exponent space sin space open parentheses 5 straight x minus tan to the power of negative 1 end exponent 5 over 2 close parentheses plus straight c 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 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 space equals space fraction numerator 1 over denominator square root of 29 end fraction sin space open parentheses 5 straight x minus space tan to the power of negative 1 end exponent 5 over 2 close parentheses plus space straight c space straight e to the power of negative 2 straight x end exponent

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