Question

Find one-parameter families of solution curves of the following differential equations:
$straight y apostrophe plus 3 straight y space equals straight e to the power of negative 2 straight x end exponent$

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Solution

The given differential equation is

straight y apostrophe plus 3 straight y space equals space straight e to the power of negative 2 straight x end exponent

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

Comparing it with

dy over dx plus Py space equals straight Q comma space we space get comma space straight P space equals space 3 comma space straight Q space equals space straight e to the power of negative 2 straight x end exponent

therefore space space space space space integral straight P space dx space equals space 3 space integral space 1 space dx space equals space 3 space straight x. space 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 space straight Q space straight e to the power of integral straight P space dx end exponent dx plus space straight c

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

or space space space straight y space straight e to the power of 3 straight x end exponent space equals space integral straight e to the power of straight x space dx plus straight c or space space space space straight y space straight e to the power of 3 straight x end exponent space equals space straight e to the power of straight x plus straight c or space space space space space space space straight y space space equals space straight e to the power of negative 2 straight x end exponent plus straight c space straight e to the power of negative 3 straight x end exponent space is space required space solution.

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