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Math Help - integration general solution

  1. #1
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    Question integration general solution

    Hi i have an integration problem. basically i am going through an example...the equation i want to find the gen solution is shown below..i have also shown the gen solution as well...but i cant understand how they got to that answers


    \xi_{yy} + (1/y)\xi_{y}=0

    The general solution is given by

    \xi = A(x)\ln(y) + B(x)

    can some one explain how they got to this general soution..thanks
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  2. #2
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    Quote Originally Posted by dopi View Post
    Hi i have an integration problem. basically i am going through an example...the equation i want to find the gen solution is shown below..i have also shown the gen solution as well...but i cant understand how they got to that answers


    \xi_{yy} + (1/y)\xi_{y}=0

    The general solution is given by

    \xi = A(x)\ln(y) + B(x)

    can some one explain how they got to this general soution..thanks
    The function \xi depends on x and y but the equation only involves y, hence it can be seen as an ordinary differential equation.

    Let x be fixed. Write f(y)=\xi(x,y). Then f is a function of 1 variable and f''(y)+\frac{1}{y}f'(y)=0 for every y. Let g=f', so that g'(y)+\frac{1}{y}g(y)=0. This is a linear first order differential equation: we know how to solve it. Since \int \frac{dy}{y}=\ln y, there exists A\in\mathbb{R} such that g(y)=A e^{-\ln y}=\frac{A}{y}. Hence, by integration, f(y)=A\ln y + B for some B\in\mathbb{R}.
    However, the "constants" A and B depend on x which was fixed before, hence we should write A(x) and B(x): \xi(x,y)=f(y)=A(x)\ln y+B(x). This is it.
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