Solve:
$log dy over dx equals ax plus by.$

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Solution

The given differential equation is

log space dy over dx space equals space ax plus by space space space space or space space space space dy over dx space equals space straight e to the power of ax plus by end exponent

dy over dx space equals space straight e to the power of ax. space straight e to the power of by.

Separating the variables, we get,

1 over straight e to the power of by dy space equals space straight e to the power of ax dx

Integrating,

integral straight e to the power of negative by end exponent dy space equals space straight e to the power of ax dx

therefore space space space space space space fraction numerator straight e to the power of negative by end exponent over denominator negative straight b end fraction space equals space straight e to the power of ax over straight a plus straight c space space space space space or space space space space minus 1 over straight b straight e to the power of negative by end exponent space equals space 1 over straight a straight e to the power of ax plus straight c

which is required solution.

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