Internal problem ID [8493]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 913.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Abel, 2nd type, class C]]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {-y^{3}-y+2 y^{2} \ln \relax (x )-\ln \relax (x )^{2} y^{3}-1+3 y \ln \relax (x )-3 \ln \relax (x )^{2} y^{2}+\ln \relax (x )^{3} y^{3}}{y x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 43
dsolve(diff(y(x),x) = -(-y(x)^3-y(x)+2*y(x)^2*ln(x)-ln(x)^2*y(x)^3-1+3*y(x)*ln(x)-3*ln(x)^2*y(x)^2+ln(x)^3*y(x)^3)/y(x)/x,y(x), singsol=all)
\[ y \relax (x ) = \frac {9}{9 \ln \relax (x )+56 \RootOf \left (-81 \left (\int _{}^{\textit {\_Z}}\frac {1}{3136 \textit {\_a}^{3}-27 \textit {\_a} +27}d \textit {\_a} \right )-\ln \relax (x )+3 c_{1}\right )-3} \]
✓ Solution by Mathematica
Time used: 0.532 (sec). Leaf size: 716
DSolve[y'[x] == (1 + y[x] - 3*Log[x]*y[x] - 2*Log[x]*y[x]^2 + 3*Log[x]^2*y[x]^2 + y[x]^3 + Log[x]^2*y[x]^3 - Log[x]^3*y[x]^3)/(x*y[x]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (2 \text {RootSum}\left [\text {$\#$1}^3 K[1]^3-\text {$\#$1}^2 K[1]^3-2 K[1]^3-3 \text {$\#$1}^2 K[1]^2+2 \text {$\#$1} K[1]^2+3 \text {$\#$1} K[1]-K[1]-1\&,\frac {\log (\log (x)-\text {$\#$1})}{3 \text {$\#$1}^2 K[1]^2-2 \text {$\#$1} K[1]^2-6 \text {$\#$1} K[1]+2 K[1]+3}\&\right ] K[1]-\frac {K[1]}{\log ^3(x) K[1]^3-\log ^2(x) K[1]^3-2 K[1]^3-3 \log ^2(x) K[1]^2+2 \log (x) K[1]^2+3 \log (x) K[1]-K[1]-1}+\frac {\text {RootSum}\left [\text {$\#$1}^3 K[1]^3-\text {$\#$1}^2 K[1]^3-2 K[1]^3-3 \text {$\#$1}^2 K[1]^2+2 \text {$\#$1} K[1]^2+3 \text {$\#$1} K[1]-K[1]-1\&,\frac {-2 \log (x) \log (\log (x)-\text {$\#$1}) \text {$\#$1}^2 K[1]^3+38 \log (\log (x)-\text {$\#$1}) \text {$\#$1}^2 K[1]^3+12 \log (x) \log (\log (x)-\text {$\#$1}) K[1]^3+4 \log (\log (x)-\text {$\#$1}) K[1]^3-36 \log (x) \log (\log (x)-\text {$\#$1}) \text {$\#$1} K[1]^3-12 \log (\log (x)-\text {$\#$1}) \text {$\#$1} K[1]^3+2 \log (\log (x)-\text {$\#$1}) \text {$\#$1}^2 K[1]^2+\text {$\#$1}^2 K[1]^2+36 \log (x) \log (\log (x)-\text {$\#$1}) K[1]^2+4 \log (x) \log (\log (x)-\text {$\#$1}) \text {$\#$1} K[1]^2-40 \log (\log (x)-\text {$\#$1}) \text {$\#$1} K[1]^2+18 \text {$\#$1} K[1]^2-6 K[1]^2-2 \log (x) \log (\log (x)-\text {$\#$1}) K[1]+2 \log (\log (x)-\text {$\#$1}) K[1]-4 \log (\log (x)-\text {$\#$1}) \text {$\#$1} K[1]-2 \text {$\#$1} K[1]-18 K[1]+2 \log (\log (x)-\text {$\#$1})+1}{\log (x) \text {$\#$1}^2 K[1]^3-19 \text {$\#$1}^2 K[1]^3+110 \log (x) K[1]^3+18 \log (x) \text {$\#$1} K[1]^3-110 \text {$\#$1} K[1]^3-2 K[1]^3-\text {$\#$1}^2 K[1]^2-18 \log (x) K[1]^2-2 \log (x) \text {$\#$1} K[1]^2+20 \text {$\#$1} K[1]^2+\log (x) K[1]+2 \text {$\#$1} K[1]-K[1]-1}\&\right ]}{K[1]}\right )dK[1]+y(x)^2 \left (-\text {RootSum}\left [\text {$\#$1}^3 y(x)^3-\text {$\#$1}^2 y(x)^3-3 \text {$\#$1}^2 y(x)^2+2 \text {$\#$1} y(x)^2+3 \text {$\#$1} y(x)-2 y(x)^3-y(x)-1\&,\frac {\log (\log (x)-\text {$\#$1})}{3 \text {$\#$1}^2 y(x)^2-2 \text {$\#$1} y(x)^2-6 \text {$\#$1} y(x)+2 y(x)+3}\&\right ]\right )-\log (x)=c_1,y(x)\right ] \]