Internal problem ID [7832]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 252.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {\left (y x^{2}-1\right ) y^{\prime }-x y^{2}+1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 1583
dsolve((x^2*y(x)-1)*diff(y(x),x)-(x*y(x)^2-1)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {63 x^{3}-\frac {63 x^{2} \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}}\right ) \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}-\frac {63 c_{1} x^{4}}{\left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}}\right ) \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}}{4 x^{2} \left (\frac {63 x^{2} \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}}\right ) \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{4 \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )}+\frac {63 c_{1} x^{4}}{4 \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}}\right ) \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}-\frac {63}{4}\right )} \\ y \relax (x ) = -\frac {63 x^{3}+\frac {63 x^{2} \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}}\right ) \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{2 \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )}+\frac {63 c_{1} x^{4}}{2 \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}}\right ) \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}-2 i \sqrt {3}\, \left (\frac {63 x^{2} \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}}\right ) \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{4 \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )}-\frac {63 c_{1} x^{4}}{4 \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}}\right ) \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}\right )}{4 x^{2} \left (-\frac {63 x^{2} \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}}\right ) \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{8 \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )}-\frac {63 c_{1} x^{4}}{8 \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}}\right ) \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}-\frac {63}{4}+\frac {i \sqrt {3}\, \left (\frac {63 x^{2} \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}}\right ) \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{4 \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )}-\frac {63 c_{1} x^{4}}{4 \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}}\right ) \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ y \relax (x ) = -\frac {63 x^{3}+\frac {63 x^{2} \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}}\right ) \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{2 \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )}+\frac {63 c_{1} x^{4}}{2 \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}}\right ) \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}+2 i \sqrt {3}\, \left (\frac {63 x^{2} \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}}\right ) \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{4 \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )}-\frac {63 c_{1} x^{4}}{4 \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}}\right ) \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}\right )}{4 x^{2} \left (-\frac {63 x^{2} \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}}\right ) \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{8 \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )}-\frac {63 c_{1} x^{4}}{8 \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}}\right ) \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}-\frac {63}{4}-\frac {i \sqrt {3}\, \left (\frac {63 x^{2} \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}}\right ) \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{4 \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )}-\frac {63 c_{1} x^{4}}{4 \left (c_{1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{x^{6} c_{1}-80 x^{6}+160 x^{3}-80}}\right ) \left (x^{6} c_{1}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 34.222 (sec). Leaf size: 476
DSolve[(x^2*y[x]-1)*y'[x]-(x*y[x]^2-1)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (2 x^3+6 c_1 \left (x^3-1\right )^2-1\right )}+1+12 c_1 (-1+3 c_1)}}{-1+6 c_1}-\frac {x^2}{\sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (2 x^3+6 c_1 \left (x^3-1\right )^2-1\right )}+1+12 c_1 (-1+3 c_1)}}+x \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (2 x^3+6 c_1 \left (x^3-1\right )^2-1\right )}+1+12 c_1 (-1+3 c_1)}}{-2+12 c_1}+\frac {\left (1+i \sqrt {3}\right ) x^2}{2 \sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (2 x^3+6 c_1 \left (x^3-1\right )^2-1\right )}+1+12 c_1 (-1+3 c_1)}}+x \\ y(x)\to -\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (2 x^3+6 c_1 \left (x^3-1\right )^2-1\right )}+1+12 c_1 (-1+3 c_1)}}{-2+12 c_1}+\frac {\left (1-i \sqrt {3}\right ) x^2}{2 \sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (2 x^3+6 c_1 \left (x^3-1\right )^2-1\right )}+1+12 c_1 (-1+3 c_1)}}+x \\ y(x)\to x \\ \end{align*}