Differential Equations

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Question

Find one-parameter families of solution curves of the following differential equation:
x y' + y = x⁴

Solution

The given differential equation is
x y' + y = x⁴or

dy over dx plus 1 over straight x straight y space equals space straight x cubed

Comparing it with

dy over dx plus Py space equals straight Q comma space we space get space straight P space equals 1 over straight x. straight Q space equals space straight x cubed

straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of integral 1 over straight x dx end exponent space equals space straight e to the power of log space straight x end exponent space equals space straight x

Solution of differential equation is

straight y space straight e to the power of integral space 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

straight y. space straight e space equals space integral straight x cubed. space straight x space dx plus straight c space space space space space space or space space space straight x space straight y space equals space integral straight x to the power of 4 dx plus straight c

xy space equals straight x to the power of 5 over 5 plus straight c space space space space space space space space space space space space space space space space space space space or space space space space space straight y space equals space straight x to the power of 4 over 5 plus straight c over straight x