Internal problem ID [8563]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 983.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x),G(x)]], _Abel]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y^{3}-3 x y^{2}+3 x^{2} y-x^{3}+x^{2}}{\left (x -1\right ) \left (x +1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 375
dsolve(diff(y(x),x) = (y(x)^3-3*x*y(x)^2+3*x^2*y(x)-x^3+x^2)/(x-1)/(x+1),y(x), singsol=all)
\[ y \relax (x ) = \frac {\sqrt {3}\, \left (\frac {1}{\left (x +1\right )^{3} \left (x -1\right )^{3}}\right )^{\frac {1}{3}} x^{2}+3 \tan \left (\RootOf \left (-9 \left (\frac {1}{\left (x +1\right )^{3} \left (x -1\right )^{3}}\right )^{\frac {2}{3}} \ln \left (\frac {x +1}{x -1}\right ) x^{4}+18 \left (\frac {1}{\left (x +1\right )^{3} \left (x -1\right )^{3}}\right )^{\frac {2}{3}} \ln \left (\frac {x +1}{x -1}\right ) x^{2}-9 \left (\frac {1}{\left (x +1\right )^{3} \left (x -1\right )^{3}}\right )^{\frac {2}{3}} \ln \left (\frac {x +1}{x -1}\right )-6 \textit {\_Z} \sqrt {3}-3 \ln \left (\frac {4}{3 \left (\tan ^{2}\left (\textit {\_Z} \right )+1\right )}\right )-2 \ln \left (\frac {27 \left (\sqrt {3}+\tan \left (\textit {\_Z} \right )\right )^{3}}{8 \left (x +1\right )^{3} \left (x -1\right )^{3} \left (\sqrt {3}\right )^{3}}\right )+2 \ln \left (\frac {1}{\left (x +1\right )^{3} \left (x -1\right )^{3}}\right )+18 c_{1}\right )\right ) \left (\frac {1}{\left (x +1\right )^{3} \left (x -1\right )^{3}}\right )^{\frac {1}{3}} x^{2}-\sqrt {3}\, \left (\frac {1}{\left (x +1\right )^{3} \left (x -1\right )^{3}}\right )^{\frac {1}{3}}-3 \tan \left (\RootOf \left (-9 \left (\frac {1}{\left (x +1\right )^{3} \left (x -1\right )^{3}}\right )^{\frac {2}{3}} \ln \left (\frac {x +1}{x -1}\right ) x^{4}+18 \left (\frac {1}{\left (x +1\right )^{3} \left (x -1\right )^{3}}\right )^{\frac {2}{3}} \ln \left (\frac {x +1}{x -1}\right ) x^{2}-9 \left (\frac {1}{\left (x +1\right )^{3} \left (x -1\right )^{3}}\right )^{\frac {2}{3}} \ln \left (\frac {x +1}{x -1}\right )-6 \textit {\_Z} \sqrt {3}-3 \ln \left (\frac {4}{3 \left (\tan ^{2}\left (\textit {\_Z} \right )+1\right )}\right )-2 \ln \left (\frac {27 \left (\sqrt {3}+\tan \left (\textit {\_Z} \right )\right )^{3}}{8 \left (x +1\right )^{3} \left (x -1\right )^{3} \left (\sqrt {3}\right )^{3}}\right )+2 \ln \left (\frac {1}{\left (x +1\right )^{3} \left (x -1\right )^{3}}\right )+18 c_{1}\right )\right ) \left (\frac {1}{\left (x +1\right )^{3} \left (x -1\right )^{3}}\right )^{\frac {1}{3}}+2 \sqrt {3}\, x}{2 \sqrt {3}} \]
✓ Solution by Mathematica
Time used: 0.509 (sec). Leaf size: 238
DSolve[y'[x] == (x^2 - x^3 + 3*x^2*y[x] - 3*x*y[x]^2 + y[x]^3)/((-1 + x)*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\text {ArcTan}\left (\frac {\frac {2 \left (\frac {3 y(x)}{x^2-1}-\frac {3 x}{x^2-1}\right )}{3 \sqrt [3]{\frac {1}{(x-1)^3 (x+1)^3}}}-1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (\frac {\frac {3 y(x)}{x^2-1}-\frac {3 x}{x^2-1}}{3 \sqrt [3]{\frac {1}{(x-1)^3 (x+1)^3}}}+1\right )-\frac {1}{6} \log \left (\frac {\left (\frac {3 y(x)}{x^2-1}-\frac {3 x}{x^2-1}\right )^2}{9 \left (\frac {1}{(x-1)^3 (x+1)^3}\right )^{2/3}}-\frac {\frac {3 y(x)}{x^2-1}-\frac {3 x}{x^2-1}}{3 \sqrt [3]{\frac {1}{(x-1)^3 (x+1)^3}}}+1\right )=\frac {1}{2} \left (\frac {1}{\left (x^2-1\right )^3}\right )^{2/3} \left (x^2-1\right )^2 (\log (1-x)-\log (x+1))+c_1,y(x)\right ] \]