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Differential Equations

Question
CBSEENMA12033173

Find one-parameter families of solution curves of the following differential equation:
x y' - y = (x+1) e-x

Solution
The given differential equation is
                      x y' - y = (x+1) e-x     or     dy over dx minus 1 over straight x straight y space equals open parentheses 1 plus 1 over straight x close parentheses straight e to the power of negative straight x end exponent
Comparing with dy over dx plus straight P space straight y space equals space straight Q comma space we space get space straight P space equals space minus 1 over straight x comma space space straight Q space equals space open parentheses 1 plus 1 over straight x close parentheses straight e to the power of negative straight x end exponent
                 integral straight P space dx space equals negative integral 1 over straight x dx space equals negative log space straight x comma space space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of log space straight x end exponent space equals space straight e to the power of log space straight x to the power of negative 1 end exponent end exponent space equals space straight x to the power of negative 1 end exponent space equals space 1 over straight x
Solution of differential equation 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 Pdx space end exponent dx plus straight c
or         straight y.1 over straight x space equals space integral open parentheses 1 plus 1 over straight x close parentheses straight e to the power of negative straight x end exponent. space 1 over straight x dx plus straight c
or          straight y over straight x equals integral open parentheses 1 over straight x plus 1 over straight x squared close parentheses space straight e to the power of negative straight x end exponent dx plus straight c
or space space straight y over straight x equals integral 1 over straight x. straight e to the power of negative straight x end exponent space dx plus integral 1 over straight x squared straight e to the power of negative straight x end exponent. space dx plus straight c
or space space space space straight y over straight x equals 1 over straight x. space fraction numerator straight e to the power of negative straight x end exponent over denominator negative 1 end fraction minus space integral fraction numerator negative 1 over denominator straight x squared end fraction. space fraction numerator straight e to the power of negative straight x end exponent over denominator negative 1 end fraction dx plus integral 1 over straight x squared straight e to the power of negative straight x end exponent dx plus straight c
or space space space space straight y over straight x equals negative 1 over xe to the power of straight x minus integral fraction numerator 1 over denominator straight x squared straight e to the power of straight x end fraction dx plus integral fraction numerator 1 over denominator straight x squared straight e to the power of straight x end fraction dx plus straight c
or space space space space straight y over straight x equals negative 1 over xe to the power of straight x plus straight c
or space space space space space straight y space equals negative straight e to the power of negative straight x end exponent plus cx

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