Question

The general solution of the differential equation e^x dy + (ye^x + 2x) dx = 0 is
x e^y + x² = C
x e^y + y² = C

y e^x + x² = C

y e^y + x² = C

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Solution

y e^x + x² = C

The given differential equation is

straight e to the power of straight x dy space plus space left parenthesis straight y space straight e to the power of straight x space plus space 2 straight x right parenthesis space dx space equals space 0

straight e to the power of straight x dy over dx plus space straight y space straight e to the power of straight x space plus space 2 space straight x space equals 0 space space space space or space space space dy over dx plus straight y plus fraction numerator 2 straight x over denominator straight e to the power of straight x end fraction space equals space 0 space space or space space dy over dx plus straight y space equals space minus 2 xe to the power of negative straight x end exponent

Comparing it with

dy over dx plus Py space equals space straight Q comma

we get, P = 1,

straight Q space equals negative 2 xe to the power of negative straight x end exponent

therefore space space space space space space space space space space integral straight P space dx space equals space integral space 1 space dx space equals space apostrophe straight x comma space straight e to the power of integral straight P space dx end exponent space equals space straight e to the power of straight x

Solution of given differential equation is

ye space 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 plus straight C space space space or space space space straight y space straight e to the power of straight x space equals space integral left parenthesis negative 2 straight x space straight e to the power of straight x right parenthesis. space straight e to the power of straight x dx plus straight C

straight y space straight e to the power of straight x space equals space minus 2 space integral space straight x space dx space plus space straight C space space space space or space space space straight y space straight e to the power of straight x space equals space minus straight x squared plus straight C

space ye to the power of straight x plus straight x squared space equals space straight C

therefore space space left parenthesis straight C right parenthesis space is space correct space answer.

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