##### 4.12.19 $$\left (1-x^2 y(x)\right ) y'(x)+x y(x)^2-1=0$$

ODE
$\left (1-x^2 y(x)\right ) y'(x)+x y(x)^2-1=0$ ODE Classiﬁcation

[_rational, [_Abel, 2nd type, class B]]

Book solution method
Homogeneous equation, special

Mathematica
cpu = 10.9911 (sec), leaf count = 501

$\left \{\left \{y(x)\to \frac {\sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (6 c_1 x^6+(2-12 c_1) x^3-1+6 c_1\right )}+1+36 c_1{}^2-12 c_1}}{-1+6 c_1}-\frac {x^2}{\sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (6 c_1 x^6+(2-12 c_1) x^3-1+6 c_1\right )}+1+36 c_1{}^2-12 c_1}}+x\right \},\left \{y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (6 c_1 x^6+(2-12 c_1) x^3-1+6 c_1\right )}+1+36 c_1{}^2-12 c_1}}{-2+12 c_1}+\frac {\left (1+i \sqrt {3}\right ) x^2}{2 \sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (6 c_1 x^6+(2-12 c_1) x^3-1+6 c_1\right )}+1+36 c_1{}^2-12 c_1}}+x\right \},\left \{y(x)\to -\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (6 c_1 x^6+(2-12 c_1) x^3-1+6 c_1\right )}+1+36 c_1{}^2-12 c_1}}{-2+12 c_1}+\frac {\left (1-i \sqrt {3}\right ) x^2}{2 \sqrt [3]{-(1-6 c_1){}^2 x^3+\sqrt {(-1+6 c_1){}^3 \left (6 c_1 x^6+(2-12 c_1) x^3-1+6 c_1\right )}+1+36 c_1{}^2-12 c_1}}+x\right \}\right \}$

Maple
cpu = 0.74 (sec), leaf count = 1583

$\left [y \left (x \right ) = -\frac {63 x^{3}-\frac {63 x^{2} \left (\textit {\_C1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}-\frac {63 \textit {\_C1} \,x^{4}}{\left (\textit {\_C1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}}{4 x^{2} \left (\frac {63 x^{2} \left (\textit {\_C1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{4 \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )}+\frac {63 \textit {\_C1} \,x^{4}}{4 \left (\textit {\_C1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}-\frac {63}{4}\right )}, y \left (x \right ) = -\frac {63 x^{3}+\frac {63 x^{2} \left (\textit {\_C1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{2 \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )}+\frac {63 \textit {\_C1} \,x^{4}}{2 \left (\textit {\_C1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}-2 i \sqrt {3}\, \left (\frac {63 x^{2} \left (\textit {\_C1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{4 \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )}-\frac {63 \textit {\_C1} \,x^{4}}{4 \left (\textit {\_C1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}\right )}{4 x^{2} \left (-\frac {63 x^{2} \left (\textit {\_C1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{8 \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )}-\frac {63 \textit {\_C1} \,x^{4}}{8 \left (\textit {\_C1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (\textit {\_C1} \,x^{6}-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 (\textit {\_C1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{4 \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )}-\frac {63 \textit {\_C1} \,x^{4}}{4 \left (\textit {\_C1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}\right )}{2}\right )}, y \left (x \right ) = -\frac {63 x^{3}+\frac {63 x^{2} \left (\textit {\_C1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{2 \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )}+\frac {63 \textit {\_C1} \,x^{4}}{2 \left (\textit {\_C1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}+2 i \sqrt {3}\, \left (\frac {63 x^{2} \left (\textit {\_C1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{4 \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )}-\frac {63 \textit {\_C1} \,x^{4}}{4 \left (\textit {\_C1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}\right )}{4 x^{2} \left (-\frac {63 x^{2} \left (\textit {\_C1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{8 \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )}-\frac {63 \textit {\_C1} \,x^{4}}{8 \left (\textit {\_C1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (\textit {\_C1} \,x^{6}-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 (\textit {\_C1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}{4 \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )}-\frac {63 \textit {\_C1} \,x^{4}}{4 \left (\textit {\_C1} \left (-1+4 \sqrt {-\frac {5 \left (x^{6}-2 x^{3}+1\right )}{\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80}}\right ) \left (\textit {\_C1} \,x^{6}-80 x^{6}+160 x^{3}-80\right )^{2}\right )^{\frac {1}{3}}}\right )}{2}\right )}\right ]$ Mathematica raw input

DSolve[-1 + x*y[x]^2 + (1 - x^2*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> x - x^2/(1 - x^3*(1 - 6*C[1])^2 - 12*C[1] + 36*C[1]^2 + Sqrt[(-1 + 6*C
[1])^3*(-1 + x^3*(2 - 12*C[1]) + 6*C[1] + 6*x^6*C[1])])^(1/3) + (1 - x^3*(1 - 6*
C[1])^2 - 12*C[1] + 36*C[1]^2 + Sqrt[(-1 + 6*C[1])^3*(-1 + x^3*(2 - 12*C[1]) + 6
*C[1] + 6*x^6*C[1])])^(1/3)/(-1 + 6*C[1])}, {y[x] -> x + ((1 + I*Sqrt[3])*x^2)/(
2*(1 - x^3*(1 - 6*C[1])^2 - 12*C[1] + 36*C[1]^2 + Sqrt[(-1 + 6*C[1])^3*(-1 + x^3
*(2 - 12*C[1]) + 6*C[1] + 6*x^6*C[1])])^(1/3)) + (I*(I + Sqrt[3])*(1 - x^3*(1 -
6*C[1])^2 - 12*C[1] + 36*C[1]^2 + Sqrt[(-1 + 6*C[1])^3*(-1 + x^3*(2 - 12*C[1]) +
 6*C[1] + 6*x^6*C[1])])^(1/3))/(-2 + 12*C[1])}, {y[x] -> x + ((1 - I*Sqrt[3])*x^
2)/(2*(1 - x^3*(1 - 6*C[1])^2 - 12*C[1] + 36*C[1]^2 + Sqrt[(-1 + 6*C[1])^3*(-1 +
 x^3*(2 - 12*C[1]) + 6*C[1] + 6*x^6*C[1])])^(1/3)) - (I*(-I + Sqrt[3])*(1 - x^3*
(1 - 6*C[1])^2 - 12*C[1] + 36*C[1]^2 + Sqrt[(-1 + 6*C[1])^3*(-1 + x^3*(2 - 12*C[
1]) + 6*C[1] + 6*x^6*C[1])])^(1/3))/(-2 + 12*C[1])}}

Maple raw input

dsolve((1-x^2*y(x))*diff(y(x),x)-1+x*y(x)^2 = 0, y(x))

Maple raw output

[y(x) = -1/4*(63*x^3-63*x^2/(_C1*x^6-80*x^6+160*x^3-80)*(_C1*(-1+4*(-5*(x^6-2*x^
3+1)/(_C1*x^6-80*x^6+160*x^3-80))^(1/2))*(_C1*x^6-80*x^6+160*x^3-80)^2)^(1/3)-63
*_C1*x^4/(_C1*(-1+4*(-5*(x^6-2*x^3+1)/(_C1*x^6-80*x^6+160*x^3-80))^(1/2))*(_C1*x
^6-80*x^6+160*x^3-80)^2)^(1/3))/x^2/(63/4*x^2/(_C1*x^6-80*x^6+160*x^3-80)*(_C1*(
-1+4*(-5*(x^6-2*x^3+1)/(_C1*x^6-80*x^6+160*x^3-80))^(1/2))*(_C1*x^6-80*x^6+160*x
^3-80)^2)^(1/3)+63/4*_C1*x^4/(_C1*(-1+4*(-5*(x^6-2*x^3+1)/(_C1*x^6-80*x^6+160*x^
3-80))^(1/2))*(_C1*x^6-80*x^6+160*x^3-80)^2)^(1/3)-63/4), y(x) = -1/4*(63*x^3+63
/2*x^2/(_C1*x^6-80*x^6+160*x^3-80)*(_C1*(-1+4*(-5*(x^6-2*x^3+1)/(_C1*x^6-80*x^6+
160*x^3-80))^(1/2))*(_C1*x^6-80*x^6+160*x^3-80)^2)^(1/3)+63/2*_C1*x^4/(_C1*(-1+4
*(-5*(x^6-2*x^3+1)/(_C1*x^6-80*x^6+160*x^3-80))^(1/2))*(_C1*x^6-80*x^6+160*x^3-8
0)^2)^(1/3)-2*I*3^(1/2)*(63/4*x^2/(_C1*x^6-80*x^6+160*x^3-80)*(_C1*(-1+4*(-5*(x^
6-2*x^3+1)/(_C1*x^6-80*x^6+160*x^3-80))^(1/2))*(_C1*x^6-80*x^6+160*x^3-80)^2)^(1
/3)-63/4*_C1*x^4/(_C1*(-1+4*(-5*(x^6-2*x^3+1)/(_C1*x^6-80*x^6+160*x^3-80))^(1/2)
)*(_C1*x^6-80*x^6+160*x^3-80)^2)^(1/3)))/x^2/(-63/8*x^2/(_C1*x^6-80*x^6+160*x^3-
80)*(_C1*(-1+4*(-5*(x^6-2*x^3+1)/(_C1*x^6-80*x^6+160*x^3-80))^(1/2))*(_C1*x^6-80
*x^6+160*x^3-80)^2)^(1/3)-63/8*_C1*x^4/(_C1*(-1+4*(-5*(x^6-2*x^3+1)/(_C1*x^6-80*
x^6+160*x^3-80))^(1/2))*(_C1*x^6-80*x^6+160*x^3-80)^2)^(1/3)-63/4+1/2*I*3^(1/2)*
(63/4*x^2/(_C1*x^6-80*x^6+160*x^3-80)*(_C1*(-1+4*(-5*(x^6-2*x^3+1)/(_C1*x^6-80*x
^6+160*x^3-80))^(1/2))*(_C1*x^6-80*x^6+160*x^3-80)^2)^(1/3)-63/4*_C1*x^4/(_C1*(-
1+4*(-5*(x^6-2*x^3+1)/(_C1*x^6-80*x^6+160*x^3-80))^(1/2))*(_C1*x^6-80*x^6+160*x^
3-80)^2)^(1/3))), y(x) = -1/4*(63*x^3+63/2*x^2/(_C1*x^6-80*x^6+160*x^3-80)*(_C1*
(-1+4*(-5*(x^6-2*x^3+1)/(_C1*x^6-80*x^6+160*x^3-80))^(1/2))*(_C1*x^6-80*x^6+160*
x^3-80)^2)^(1/3)+63/2*_C1*x^4/(_C1*(-1+4*(-5*(x^6-2*x^3+1)/(_C1*x^6-80*x^6+160*x
^3-80))^(1/2))*(_C1*x^6-80*x^6+160*x^3-80)^2)^(1/3)+2*I*3^(1/2)*(63/4*x^2/(_C1*x
^6-80*x^6+160*x^3-80)*(_C1*(-1+4*(-5*(x^6-2*x^3+1)/(_C1*x^6-80*x^6+160*x^3-80))^
(1/2))*(_C1*x^6-80*x^6+160*x^3-80)^2)^(1/3)-63/4*_C1*x^4/(_C1*(-1+4*(-5*(x^6-2*x
^3+1)/(_C1*x^6-80*x^6+160*x^3-80))^(1/2))*(_C1*x^6-80*x^6+160*x^3-80)^2)^(1/3)))
/x^2/(-63/8*x^2/(_C1*x^6-80*x^6+160*x^3-80)*(_C1*(-1+4*(-5*(x^6-2*x^3+1)/(_C1*x^
6-80*x^6+160*x^3-80))^(1/2))*(_C1*x^6-80*x^6+160*x^3-80)^2)^(1/3)-63/8*_C1*x^4/(
_C1*(-1+4*(-5*(x^6-2*x^3+1)/(_C1*x^6-80*x^6+160*x^3-80))^(1/2))*(_C1*x^6-80*x^6+
160*x^3-80)^2)^(1/3)-63/4-1/2*I*3^(1/2)*(63/4*x^2/(_C1*x^6-80*x^6+160*x^3-80)*(_
C1*(-1+4*(-5*(x^6-2*x^3+1)/(_C1*x^6-80*x^6+160*x^3-80))^(1/2))*(_C1*x^6-80*x^6+1
60*x^3-80)^2)^(1/3)-63/4*_C1*x^4/(_C1*(-1+4*(-5*(x^6-2*x^3+1)/(_C1*x^6-80*x^6+16
0*x^3-80))^(1/2))*(_C1*x^6-80*x^6+160*x^3-80)^2)^(1/3)))]