Question

Solve $dy over dx plus straight y space equals space 1 comma space space space straight y not equal to 1$

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Solution

The given differential equation is

dy over dx plus straight y space equals space 1 space space space or space space space dy over dx equals space 1 minus straight y

Separating the variables, we get,

fraction numerator 1 over denominator 1 minus straight y end fraction dy space equals space dx

Integrating,

integral fraction numerator 1 over denominator 1 minus straight y end fraction dy space equals space integral 1 space dx

therefore space space space space space space fraction numerator log open vertical bar 1 minus straight y close vertical bar over denominator negative 1 end fraction space equals space straight x plus straight c apostrophe

therefore space space space space space log space open vertical bar 1 minus straight y close vertical bar space equals space minus straight x minus straight c apostrophe

therefore space space space space space space space space open vertical bar 1 minus straight y close vertical bar space equals space straight e to the power of negative straight x minus straight c apostrophe end exponent space space space or space space space open vertical bar 1 minus straight y close vertical bar space equals space straight e to the power of negative straight x end exponent. space straight e to the power of negative straight c apostrophe end exponent therefore space space space space space space space space space space space 1 minus straight y space equals space plus-or-minus straight e to the power of negative straight e apostrophe end exponent space straight e to the power of negative straight x end exponent therefore space space space space space space space space space space space space space 1 minus straight y space equals space minus ce to the power of negative straight x end exponent space space space space space space space space space space space space space space space space space space space space space space space where space minus straight c space equals space plus-or-minus straight e to the power of negative straight x end exponent therefore space space space space space space space space space space space straight y space equals space 1 plus ce to the power of negative straight x end exponent space space is space the space required space solution. space

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