60.2.189 problem 765
Internal
problem
ID
[10776]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
765
Date
solved
:
Monday, January 27, 2025 at 09:43:36 PM
CAS
classification
:
[_Bernoulli]
\begin{align*} y^{\prime }&=\frac {y \left (-1-\ln \left (\frac {\left (x -1\right ) \left (1+x \right )}{x}\right )+\ln \left (\frac {\left (x -1\right ) \left (1+x \right )}{x}\right ) x y\right )}{x} \end{align*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 53
dsolve(diff(y(x),x) = y(x)*(-1-ln((x-1)*(x+1)/x)+ln((x-1)*(x+1)/x)*x*y(x))/x,y(x), singsol=all)
\[
y = \frac {{\mathrm e}^{\operatorname {dilog}\left (x +1\right )}}{x \left (c_{1} {\mathrm e}^{\operatorname {dilog}\left (x \right )+\frac {\ln \left (x \right )^{2}}{2}} \left (\frac {x^{2}-1}{x}\right )^{\ln \left (x \right )} \left (x +1\right )^{-\ln \left (x \right )}+{\mathrm e}^{\operatorname {dilog}\left (x +1\right )}\right )}
\]
✓ Solution by Mathematica
Time used: 0.734 (sec). Leaf size: 240
DSolve[D[y[x],x] == (y[x]*(-1 - Log[((-1 + x)*(1 + x))/x] + x*Log[((-1 + x)*(1 + x))/x]*y[x]))/x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {e^{\operatorname {PolyLog}(2,-x)-\operatorname {PolyLog}(2,1-x)} x^{-\frac {\log (x)}{2}+\log (x+1)-\log \left (x-\frac {1}{x}\right )-1}}{-\int _1^xe^{\operatorname {PolyLog}(2,-K[1])-\operatorname {PolyLog}(2,1-K[1])} K[1]^{-\frac {1}{2} \log (K[1])+\log (K[1]+1)-\log \left (K[1]-\frac {1}{K[1]}\right )-1} \log \left (K[1]-\frac {1}{K[1]}\right )dK[1]+c_1} \\
y(x)\to 0 \\
y(x)\to -\frac {e^{\operatorname {PolyLog}(2,-x)-\operatorname {PolyLog}(2,1-x)} x^{-\frac {\log (x)}{2}+\log (x+1)-\log \left (x-\frac {1}{x}\right )-1}}{\int _1^xe^{\operatorname {PolyLog}(2,-K[1])-\operatorname {PolyLog}(2,1-K[1])} K[1]^{-\frac {1}{2} \log (K[1])+\log (K[1]+1)-\log \left (K[1]-\frac {1}{K[1]}\right )-1} \log \left (K[1]-\frac {1}{K[1]}\right )dK[1]} \\
\end{align*}