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Math Help - second order differential equation for IVP

  1. #1
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    Question second order differential equation for IVP

    hi guys, Im wondering how to solve this problem


    ''Consider the following initial value problem:




    By observing that the first two terms on the right-hand side of the equation form a total derivative of a function, find the analytical solution of the problem.''

    thanks
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  2. #2
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    Quote Originally Posted by hazeleyes View Post
    hi guys, Im wondering how to solve this problem


    ''Consider the following initial value problem:




    By observing that the first two terms on the right-hand side of the equation form a total derivative of a function, find the analytical solution of the problem.''

    thanks
    What the hint is tell you is notice that

     \frac{2}{x}y'-\frac{2}{x^2}y=\frac{d}{dx}\left( \frac{2}{x}y\right)

    So using this identity we get that

    \frac{d}{dx}(y')=\frac{d}{dx}\left( \frac{2}{x}y \right)-\frac{1}{x^2}

    Now move everything to the left hand side

    \frac{d}{dx}\left(y'- \frac{2}{x}y \right)=-\frac{1}{x^2}

    Now just integrate to get a first order ode

    Can you finish from here?
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