Internal problem ID [8315]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 735.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(y)]], [_Abel, 2nd type, class C]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\left (2 y \ln \relax (x )-1\right )^{3}}{\left (-1+2 y \ln \relax (x )-y\right ) x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 104
dsolve(diff(y(x),x) = (-1+2*y(x)*ln(x))^3/(-1+2*y(x)*ln(x)-y(x))/x,y(x), singsol=all)
\[ y \relax (x ) = \frac {71 \RootOf \left (-82944 \left (\int _{}^{\textit {\_Z}}\frac {1}{5041 \textit {\_a}^{3}-27648 \textit {\_a} +27648}d \textit {\_a} \right )-16 \ln \relax (x )+3 c_{1}\right )-120}{142 \ln \relax (x ) \RootOf \left (-82944 \left (\int _{}^{\textit {\_Z}}\frac {1}{5041 \textit {\_a}^{3}-27648 \textit {\_a} +27648}d \textit {\_a} \right )-16 \ln \relax (x )+3 c_{1}\right )-240 \ln \relax (x )-71 \RootOf \left (-82944 \left (\int _{}^{\textit {\_Z}}\frac {1}{5041 \textit {\_a}^{3}-27648 \textit {\_a} +27648}d \textit {\_a} \right )-16 \ln \relax (x )+3 c_{1}\right )+48} \]
✓ Solution by Mathematica
Time used: 0.837 (sec). Leaf size: 573
DSolve[y'[x] == (-1 + 2*Log[x]*y[x])^3/(x*(-1 - y[x] + 2*Log[x]*y[x])),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {2 (2 \log (x) K[1]-K[1]-1)}{8 \log ^3(x) K[1]^3+4 \log (x) K[1]^3-2 K[1]^3-12 \log ^2(x) K[1]^2-2 K[1]^2+6 \log (x) K[1]-1}+2 \text {RootSum}\left [2 K[1]^3-2 \text {$\#$1} K[1]^2-\text {$\#$1}^3\&,\frac {K[1] \log (2 K[1] \log (x)-\text {$\#$1}-1)-\log (2 K[1] \log (x)-\text {$\#$1}-1) \text {$\#$1}}{2 K[1]^2+3 \text {$\#$1}^2}\&\right ]+\frac {\text {RootSum}\left [2 K[1]^3-2 \text {$\#$1} K[1]^2-\text {$\#$1}^3\&,\frac {16 \log (x) K[1]^3-16 \log (x) \log (2 K[1] \log (x)-\text {$\#$1}-1) K[1]^3-24 \log (2 K[1] \log (x)-\text {$\#$1}-1) K[1]^3+24 K[1]^3+8 \log (2 K[1] \log (x)-\text {$\#$1}-1) K[1]^2+2 \log (x) \text {$\#$1} K[1]^2-2 \log (x) \log (2 K[1] \log (x)-\text {$\#$1}-1) \text {$\#$1} K[1]^2+32 \log (2 K[1] \log (x)-\text {$\#$1}-1) \text {$\#$1} K[1]^2-32 \text {$\#$1} K[1]^2-24 \log (x) \text {$\#$1}^2 K[1]+24 \log (x) \log (2 K[1] \log (x)-\text {$\#$1}-1) \text {$\#$1}^2 K[1]+\log (2 K[1] \log (x)-\text {$\#$1}-1) \text {$\#$1}^2 K[1]-\text {$\#$1}^2 K[1]+\log (2 K[1] \log (x)-\text {$\#$1}-1) \text {$\#$1} K[1]-12 \log (2 K[1] \log (x)-\text {$\#$1}-1) \text {$\#$1}^2}{58 \log (x) K[1]^3-18 K[1]^3-54 \log (x) \text {$\#$1} K[1]^2-11 \text {$\#$1} K[1]^2-29 K[1]^2+18 \log (x) \text {$\#$1}^2 K[1]+27 \text {$\#$1}^2 K[1]+27 \text {$\#$1} K[1]-9 \text {$\#$1}^2}\&\right ]}{K[1]}\right )dK[1]-2 \left (y(x) \text {RootSum}\left [-\text {$\#$1}^3-2 \text {$\#$1} y(x)^2+2 y(x)^3\&,\frac {y(x) \log (-\text {$\#$1}+2 y(x) \log (x)-1)-\text {$\#$1} \log (-\text {$\#$1}+2 y(x) \log (x)-1)}{3 \text {$\#$1}^2+2 y(x)^2}\&\right ]+\log (x)\right )=c_1,y(x)\right ] \]