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Differential Equations

Question

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Solve $dy over dx plus straight y over straight x space equals space straight e to the power of straight x comma space space left parenthesis straight x greater than 0 right parenthesis.$

Solution

The given differential equation is

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

Comparing it with

dy over dx plus Py space equals space straight Q comma

we get,

straight P space equals space 1 over straight x comma space space straight Q space equals space straight e to the power of straight x

integral Pdx space equals integral 1 over straight x dx space equals space log space straight x comma space space space space space straight e to the power of integral straight p space dx end exponent space equals space straight e to the power of logx space equals space straight x

therefore space space space solution space of space differential space equation space is

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

or

straight x space straight y space equals space straight x space straight e to the power of straight x space space minus space integral 1. space straight e to the power of straight x space dx plus space straight c space or space straight x space straight y space equals space straight x space straight e to the power of straight x minus straight e to the power of straight x plus straight c

or

straight y equals fraction numerator left parenthesis straight x minus 1 right parenthesis space straight e to the power of straight x over denominator straight x end fraction plus straight c over straight x

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