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Differential Equations

Question

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Solve:
$dy over dx plus 3 straight y space equals space cos space 2 straight x$

Solution

The given differential equation is

dy over dx plus 3 straight y space equals space cos space 2 straight x

Comparing it with

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

we get,
P = 3, Q = cos 2x

integral straight P space dx space equals space integral 3 space dx space equals space 3 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 3 straight x end exponent

Solution of given differential equation is

straight y. space straight e to the power of integral straight P space dx end exponent space equals space integral Qe 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 3 straight x end exponent space equals space integral straight e to the power of 3 straight x end exponent space cos space 2 straight x space dx plus straight c

or

ye to the power of 3 straight x end exponent space equals space fraction numerator 1 over denominator square root of 9 plus 4 end root end fraction space straight e to the power of 3 straight x end exponent space cos space open parentheses 2 straight x minus tan to the power of negative 1 end exponent 2 over 3 close parentheses plus straight c

or

ye to the power of 3 straight x end exponent space equals space fraction numerator 1 over denominator square root of 13 end fraction straight e to the power of 3 straight x end exponent space cos space open parentheses 2 straight x minus tan to the power of negative 1 end exponent 2 over 3 close parentheses plus straight c

therefore

straight y space equals fraction numerator 1 over denominator square root of 13 end fraction cos space open parentheses 2 straight x minus tan to the power of negative 1 end exponent 2 over 3 close parentheses plus space straight c space straight e to the power of negative 3 straight x end exponent

is the required solution.

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