Internal problem ID [8325]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 745.
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 (-1+y \ln \relax (x )\right )^{3}}{\left (-1+y \ln \relax (x )-y\right ) x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 104
dsolve(diff(y(x),x) = (-1+y(x)*ln(x))^3/(-1+y(x)*ln(x)-y(x))/x,y(x), singsol=all)
\[ y \relax (x ) = \frac {47 \RootOf \left (-27783 \left (\int _{}^{\textit {\_Z}}\frac {1}{2209 \textit {\_a}^{3}-9261 \textit {\_a} +9261}d \textit {\_a} \right )-7 \ln \relax (x )+3 c_{1}\right )-84}{47 \ln \relax (x ) \RootOf \left (-27783 \left (\int _{}^{\textit {\_Z}}\frac {1}{2209 \textit {\_a}^{3}-9261 \textit {\_a} +9261}d \textit {\_a} \right )-7 \ln \relax (x )+3 c_{1}\right )-84 \ln \relax (x )-47 \RootOf \left (-27783 \left (\int _{}^{\textit {\_Z}}\frac {1}{2209 \textit {\_a}^{3}-9261 \textit {\_a} +9261}d \textit {\_a} \right )-7 \ln \relax (x )+3 c_{1}\right )+21} \]
✓ Solution by Mathematica
Time used: 0.828 (sec). Leaf size: 546
DSolve[y'[x] == (-1 + Log[x]*y[x])^3/(x*(-1 - y[x] + Log[x]*y[x])),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {\log (x) K[1]-K[1]-1}{\log ^3(x) K[1]^3+\log (x) K[1]^3-K[1]^3-3 \log ^2(x) K[1]^2-K[1]^2+3 \log (x) K[1]-1}+\text {RootSum}\left [K[1]^3-\text {$\#$1} K[1]^2-\text {$\#$1}^3\&,\frac {K[1] \log (K[1] \log (x)-\text {$\#$1}-1)-\log (K[1] \log (x)-\text {$\#$1}-1) \text {$\#$1}}{K[1]^2+3 \text {$\#$1}^2}\&\right ]+\frac {\text {RootSum}\left [K[1]^3-\text {$\#$1} K[1]^2-\text {$\#$1}^3\&,\frac {4 \log (x) K[1]^3-4 \log (x) \log (K[1] \log (x)-\text {$\#$1}-1) K[1]^3-12 \log (K[1] \log (x)-\text {$\#$1}-1) K[1]^3+12 K[1]^3+4 \log (K[1] \log (x)-\text {$\#$1}-1) K[1]^2+5 \log (x) \text {$\#$1} K[1]^2-5 \log (x) \log (K[1] \log (x)-\text {$\#$1}-1) \text {$\#$1} K[1]^2+16 \log (K[1] \log (x)-\text {$\#$1}-1) \text {$\#$1} K[1]^2-16 \text {$\#$1} K[1]^2-12 \log (x) \text {$\#$1}^2 K[1]+12 \log (x) \log (K[1] \log (x)-\text {$\#$1}-1) \text {$\#$1}^2 K[1]+5 \log (K[1] \log (x)-\text {$\#$1}-1) \text {$\#$1}^2 K[1]-5 \text {$\#$1}^2 K[1]+5 \log (K[1] \log (x)-\text {$\#$1}-1) \text {$\#$1} K[1]-12 \log (K[1] \log (x)-\text {$\#$1}-1) \text {$\#$1}^2}{28 \log (x) K[1]^3-9 K[1]^3-27 \log (x) \text {$\#$1} K[1]^2-19 \text {$\#$1} K[1]^2-28 K[1]^2+9 \log (x) \text {$\#$1}^2 K[1]+27 \text {$\#$1}^2 K[1]+27 \text {$\#$1} K[1]-9 \text {$\#$1}^2}\&\right ]}{K[1]}\right )dK[1]-y(x) \text {RootSum}\left [-\text {$\#$1}^3-\text {$\#$1} y(x)^2+y(x)^3\&,\frac {y(x) \log (-\text {$\#$1}+y(x) \log (x)-1)-\text {$\#$1} \log (-\text {$\#$1}+y(x) \log (x)-1)}{3 \text {$\#$1}^2+y(x)^2}\&\right ]-\log (x)=c_1,y(x)\right ] \]