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11 June, 09:47

Find the general solution of the equation:

x dy/dx + y = xe^x

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  1. 11 June, 12:32
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    Step-by-step explanation:

    x * (dy/dx) + y = x * (dy/dx) + 1*y=x * (dx/dy) + dx/dx*y = (d (xy) / dx)

    d (xy) / dx = x^ex

    d (xy) = xe^x*dx

    We integrate both parts with their respective variables.

    Integral of d (xy) is xy+c.

    Integral of xe^x*dx is equal to xe^x - e^x+c.

    We have to integrate by parts:

    Integral (x*e^x) = x*e^x - integral ((x) 'e^x) = x*e^x - integral (e^x) = x*e^x-e^x.

    We have that xy + c = x*e^x - e^x + c1, where c and c1 are non defined real constants.

    So we get that xy = x*e^x - e^x + c2, where c2 is a real constant.
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