First change to standard form of first order linear differential equation and then by finding the appropriate integrating factor find a part

Question

First change to standard form of first order linear differential equation and then by finding the appropriate integrating factor find a particular solution for the initial value problem:xy’ + y = lnx ; y(e)=1

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Emery 7 days 2021-10-11T15:28:06+00:00 1 Answer 0

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    2021-10-11T15:29:17+00:00

    Answer:

    Step-by-step explanation:

    Given is a Differential equation as

    xy' + y = lnx ; y(e)=1

    To bring it to linear form we can divide the full equation by x

    y'+\frac{y}{x} =\frac{ln x}{x}

    This is of the form

    y’+p(x) *y = q(x)

    p(x) = 1/x

    So find

    e^{\int\limits{\frac{1}{x} } \, dx } = e^{ln x} = x

    Solution is

    xy = \int  {x*lnx /x } \, dx =xln x -x +C

    Use the initial value as y(e) =1

    e= eln e -e+C\\C=e

    So solution is

    xy =xln x -x+e

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