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Differential equation/Integration problem

  1. Dec 13, 2014 #1
    I have the following problem

    \begin{equation}
    \frac{\mathrm{arcsinh}\left(y\right)}{y}\frac{dy}{dx}=B\end{equation}


    where B is constant. To solve the problem I separated the
    variables and obtained

    \begin{equation}
    \int\frac{\mathrm{arcsinh}\left(y\right)}{y}dy=B \int dx\end{equation}


    I used Wolfram alpha to integrate the LHS and obtained an expression
    which did not work for some reason. To check this I tried to do this
    from first principles but the attempts led to dead end. I also could
    not find such an integral in standard integral tables. Can someone
    suggest a solution method to the problem or show me how to integrate
    the LHS from first principles or prove Wolfram is right or wrong.
     
  2. jcsd
  3. Dec 13, 2014 #2

    Astronuc

    User Avatar

    Staff: Mentor

    Apparently the solution for ## \int\frac{\mathrm{sinh^{-1}}\left(y\right)}{y}dy ## is a series.

    For y2 < 1, the solution is (Integral 731.1 from H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, NY, NY 1961.

    ##y - \frac{1}{2\cdot 3\cdot{3}} y^3 + \frac{1\cdot{3}}{2\cdot{4}\cdot{5}\cdot{5}} y^5 - \frac{1\cdot{3}\cdot{5}}{2\cdot{4}\cdot{6}\cdot{7}\cdot{7}} y^7 + . . .##

    for a more general solution, let y = x/a, and for y > 1 or < 1 the solution is somewhat different with even exponents.
    ## \int\frac{\mathrm{sinh^{-1}}\left(y\right)}{y^m}dy ## can be solved analytical for some m, at least for m = 2, 3
     
  4. Dec 14, 2014 #3
    Thanks Astronuc for your useful remark!
    I solved the problem by testing Wolfram numerically using numerical integration. I noticed that Wolfram expression produces large errors in some cases.
    Replacing Wolfram expression with numerical integration I obtained almost perfect results.
     
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