Internal problem ID [9057]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 722.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class C`]]
\[ \boxed {y^{\prime }+\frac {y^{3}}{\left (-1+2 y \ln \left (x \right )-y\right ) x}=0} \]
✓ Solution by Maple
Time used: 0.0 (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)
\[ y \left (x \right ) = \frac {{\mathrm e}^{\operatorname {RootOf}\left ({\mathrm e}^{\textit {\_Z}} \ln \left (2\right )-{\mathrm e}^{\textit {\_Z}} \ln \left (\frac {{\mathrm e}^{\textit {\_Z}}+2}{x^{4}}\right )+3 c_{1} {\mathrm e}^{\textit {\_Z}}+{\mathrm e}^{\textit {\_Z}} \textit {\_Z} +2\right )}}{1+\left (2 \ln \left (x \right )-1\right ) {\mathrm e}^{\operatorname {RootOf}\left ({\mathrm e}^{\textit {\_Z}} \ln \left (2\right )-{\mathrm e}^{\textit {\_Z}} \ln \left (\frac {{\mathrm e}^{\textit {\_Z}}+2}{x^{4}}\right )+3 c_{1} {\mathrm e}^{\textit {\_Z}}+{\mathrm e}^{\textit {\_Z}} \textit {\_Z} +2\right )}} \]
✓ Solution by Mathematica
Time used: 17.707 (sec). Leaf size: 490
DSolve[y'[x] == -(y[x]^3/(x*(-1 - y[x] + 2*Log[x]*y[x]))),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {\sqrt [3]{-2} \left ((-2)^{2/3}-\frac {(1-2 \log (x))^2 \left (-\frac {1}{(2 \log (x)-1)^3}\right )^{2/3} (y(x) (5-4 \log (x))+2)}{2 \sqrt [3]{2} (y(x) (2 \log (x)-1)-1)}\right ) \left (\frac {y(x) (4 \log (x)-5)-2}{\sqrt [3]{2} \sqrt [3]{-\frac {1}{(2 \log (x)-1)^3}} (2 \log (x)-1) (y(x) (2 \log (x)-1)-1)}+(-2)^{2/3}\right ) \left (\log \left (\frac {y(x) (5-4 \log (x))+2}{\sqrt [3]{2} \sqrt [3]{-\frac {1}{(2 \log (x)-1)^3}} (2 \log (x)-1) (y(x) (2 \log (x)-1)-1)}+2 (-2)^{2/3}\right ) \left (\frac {\sqrt [3]{-1} \left (-\frac {1}{(2 \log (x)-1)^3}\right )^{2/3} (1-2 \log (x))^2 (y(x) (4 \log (x)-5)-2)}{y(x) (4 \log (x)-2)-2}+1\right )-\log \left (\frac {y(x) (4 \log (x)-5)-2}{\sqrt [3]{2} \sqrt [3]{-\frac {1}{(2 \log (x)-1)^3}} (2 \log (x)-1) (y(x) (2 \log (x)-1)-1)}+(-2)^{2/3}\right ) \left (\frac {\sqrt [3]{-1} \left (-\frac {1}{(2 \log (x)-1)^3}\right )^{2/3} (1-2 \log (x))^2 (y(x) (4 \log (x)-5)-2)}{y(x) (4 \log (x)-2)-2}+1\right )+3\right )}{9 \left (\frac {(y(x) (4 \log (x)-5)-2)^3}{8 (y(x) (2 \log (x)-1)-1)^3}+\frac {3 \sqrt [3]{-1} (y(x) (4 \log (x)-5)-2)}{2 (1-2 \log (x))^4 \left (-\frac {1}{(2 \log (x)-1)^3}\right )^{4/3} (y(x) (2 \log (x)-1)-1)}+2\right )}=\frac {4}{9} 2^{2/3} \log (x) \left (-\frac {1}{(2 \log (x)-1)^3}\right )^{2/3} (1-2 \log (x))^2+c_1,y(x)\right ] \]