60.2.190 problem 766
Internal
problem
ID
[10777]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
766
Date
solved
:
Monday, January 27, 2025 at 09:43:54 PM
CAS
classification
:
[_Bernoulli]
\begin{align*} y^{\prime }&=\frac {y \left (-\ln \left (x \right )-x \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^{2} y\right )}{x \ln \left (x \right )} \end{align*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 85
dsolve(diff(y(x),x) = y(x)*(-ln(x)-x*ln((x-1)*(x+1)/x)+ln((x-1)*(x+1)/x)*x^2*y(x))/x/ln(x),y(x), singsol=all)
\[
y = \frac {{\mathrm e}^{-\int \frac {x \ln \left (\frac {x^{2}-1}{x}\right )+\ln \left (x \right )}{\ln \left (x \right ) x}d x}}{-\int \frac {{\mathrm e}^{-\int \frac {x \ln \left (\frac {x^{2}-1}{x}\right )+\ln \left (x \right )}{\ln \left (x \right ) x}d x} x \ln \left (\frac {x^{2}-1}{x}\right )}{\ln \left (x \right )}d x +c_{1}}
\]
✓ Solution by Mathematica
Time used: 0.582 (sec). Leaf size: 210
DSolve[D[y[x],x] == (y[x]*(-Log[x] - x*Log[((-1 + x)*(1 + x))/x] + x^2*Log[((-1 + x)*(1 + x))/x]*y[x]))/(x*Log[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\exp \left (\int _1^x\left (-\frac {\log \left (K[1]-\frac {1}{K[1]}\right )}{\log (K[1])}-\frac {1}{K[1]}\right )dK[1]\right )}{-\int _1^x\frac {\exp \left (\int _1^{K[2]}\left (-\frac {\log \left (K[1]-\frac {1}{K[1]}\right )}{\log (K[1])}-\frac {1}{K[1]}\right )dK[1]\right ) K[2] \log \left (K[2]-\frac {1}{K[2]}\right )}{\log (K[2])}dK[2]+c_1} \\
y(x)\to 0 \\
y(x)\to -\frac {\exp \left (\int _1^x\left (-\frac {\log \left (K[1]-\frac {1}{K[1]}\right )}{\log (K[1])}-\frac {1}{K[1]}\right )dK[1]\right )}{\int _1^x\frac {\exp \left (\int _1^{K[2]}\left (-\frac {\log \left (K[1]-\frac {1}{K[1]}\right )}{\log (K[1])}-\frac {1}{K[1]}\right )dK[1]\right ) K[2] \log \left (K[2]-\frac {1}{K[2]}\right )}{\log (K[2])}dK[2]} \\
\end{align*}