60.2.146 problem 722

Internal problem ID [10733]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 722
Date solved : Monday, January 27, 2025 at 09:36:33 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=-\frac {y^{3}}{\left (-1+2 y \ln \left (x \right )-y\right ) x} \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 78

dsolve(diff(y(x),x) = -y(x)^3/(-1+2*y(x)*ln(x)-y(x))/x,y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}} \ln \left (\frac {{\mathrm e}^{\textit {\_Z}}+2}{x^{4}}\right )+{\mathrm e}^{\textit {\_Z}} \ln \left (2\right )+3 c_{1} {\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+2\right )}}{1+\left (2 \ln \left (x \right )-1\right ) {\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}} \ln \left (\frac {{\mathrm e}^{\textit {\_Z}}+2}{x^{4}}\right )+{\mathrm e}^{\textit {\_Z}} \ln \left (2\right )+3 c_{1} {\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+2\right )}} \]

✓ Solution by Mathematica

Time used: 0.440 (sec). Leaf size: 122

DSolve[D[y[x],x] == -(y[x]^3/(x*(-1 - y[x] + 2*Log[x]*y[x]))),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{-\frac {(1-2 \log (x))^2 \left (-\frac {1}{(2 \log (x)-1)^3}\right )^{2/3} ((5-4 \log (x)) y(x)+2)}{2 \sqrt [3]{2} ((2 \log (x)-1) y(x)-1)}}\frac {2}{2 K[1]^3+3 \sqrt [3]{-2} K[1]+2}dK[1]=\frac {4}{9} 2^{2/3} \log (x) \left (-\frac {1}{(2 \log (x)-1)^3}\right )^{2/3} (1-2 \log (x))^2+c_1,y(x)\right ] \]