# Differential equation/Integration problem

## Main Question or Discussion Point

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.

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Astronuc
Staff Emeritus
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

• JulieK
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.