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Vector Algebra

Question

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Solve:
$dy over dx space equals straight e to the power of straight x plus straight y end exponent plus straight x squared space straight e to the power of straight y.$

Solution

dy over dx space equals straight e to the power of straight x plus straight y end exponent plus straight x squared straight e to the power of straight y space space space rightwards double arrow space space space dy over dx equals space straight e to the power of straight x. space space straight e to the power of straight y plus space straight x squared space straight e to the power of straight y space space space space rightwards double arrow space space dy over dx equals space left parenthesis straight e to the power of straight x plus straight x squared right parenthesis space straight e to the power of straight y

Separating the variables, we get,

dy over straight e to the power of straight y space equals space left parenthesis straight e to the power of straight x plus straight x squared right parenthesis space dx

therefore space space space space integral straight e to the power of negative straight y end exponent dy space equals space integral left parenthesis straight e to the power of straight x plus straight x squared right parenthesis space dx space space space rightwards double arrow space space space space fraction numerator straight e to the power of negative straight y end exponent over denominator negative 1 end fraction space equals straight e to the power of straight x plus straight x cubed over 3 plus straight c

or

straight e to the power of negative straight y end exponent plus straight e to the power of straight x plus straight x cubed over 3 plus straight c space equals space 0

which is the required solution.

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