Find the general solution of each of the following equations:
$dy over dx plus 2 straight x equals space straight e to the power of 3 straight x end exponent$

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Solution

The given differential equation is

dy over dx plus 2 straight x space equals space straight e to the power of 3 straight x end exponent space space space space or space space space space dy over dx space equals space straight e to the power of 3 straight x end exponent space minus space 2 straight x

Separating the variables, we get,

dy space equals space left parenthesis straight e to the power of 3 straight x end exponent minus 2 straight x right parenthesis space dx

Integrating,

straight y space equals space straight e to the power of 3 straight x end exponent over 3 minus 2 straight x squared over 2 space equals space straight c space space space or space space space space straight y space equals space 1 third space straight e to the power of 3 straight x end exponent space minus space straight x squared plus straight c comma space is space the space required space solution.

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