Differential Equations

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Question

Find one-parameter families of solution curves of the following differential equation:
y'+3 y = e^mx(m: a given real number)

Solution

The given differential equation is

dy over dx plus 3 straight y space equals space straight e to the power of mx

Comparing it with

dy over dx plus Py space equals space straight Q comma space space we space get space straight P space equals space 3 comma space space straight Q space equals space straight e to the power of mx

therefore space space space space space space space space space straight e to the power of integral space straight p space dx end exponent space equals straight e to the power of 3 space integral space dx end exponent space space equals space straight e to the power of 3 straight x end exponent therefore space space space space solution space is

straight y space straight e to the power of integral straight P space dx end exponent space equals integral straight Q space straight e to the power of integral straight P space dx end exponent dx plus straight c

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

straight y space straight e to the power of 3 straight x end exponent space equals space integral straight e to the power of left parenthesis straight m plus 3 right parenthesis straight x end exponent dx space equals space straight c

straight y space straight e to the power of 3 straight x end exponent space equals space fraction numerator straight e to the power of left parenthesis straight m plus 3 right parenthesis straight x end exponent over denominator straight m plus 3 end fraction plus straight c comma space space space if space straight m space plus space 3 space not equal to space 0

straight m plus 3 space equals space 0 comma space space space space space straight i. straight e. comma space straight m space equals negative 3 comma space then space solution space is space

space space space straight y space straight e to the power of 3 straight x end exponent space equals space integral 1 space dx plus straight c

straight y space straight e to the power of 3 straight x end exponent space equals space straight x plus straight c