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 = 15.0125 (sec), leaf count = 738

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

Maple
cpu = 0.044 (sec), leaf count = 101

\[ \left \{ {\frac {7}{9}\ln \left ({\frac {-63\,{x}^{3}+63}{4\,{x}^{2}y \relax (x ) -4}} \right ) }-{\frac {7}{6}\ln \left (-63\,{\frac {{x}^{2} \left (x-y \relax (x ) \right ) }{{x}^{2}y \relax (x ) -1}} \right ) }+{\frac {7}{18}\ln \left ({\frac {-63\,{x}^{3}+189\,{x}^{2}y \relax (x ) -126}{5\,{x}^{2}y \relax (x ) -5}} \right ) }-{\frac {7\,\ln \left (-1+x \right ) }{9}}-{\frac {7\,\ln \left ({x}^{2}+x+1 \right ) }{9}}+{\frac {7\,\ln \relax (x ) }{3}}-{\it \_C1}=0 \right \} \] 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^2*(1 - 6*C[1]) + x*(-1 + 6*C[1])*(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])])^(2/3))/((-1 + 6*C[1])*(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))}, {y[x] -> ((1 + I*Sqrt[3])*x^2*(-1 + 6*C[1
]) + 2*x*(-1 + 6*C[1])*(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])])^(2/3))/(2*(-1 + 6*C[1])*(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))}, {y[x] -> ((1 - I*Sqrt[3])*x^2*(-1 + 6*C[1]) + 
2*x*(-1 + 6*C[1])*(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])])^(2/3))/(2*(-1 + 6*C[1])*(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))}}

Maple raw input

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

Maple raw output

7/9*ln((-63*x^3+63)/(4*x^2*y(x)-4))-7/6*ln(-63*x^2*(x-y(x))/(x^2*y(x)-1))+7/18*l
n((-63*x^3+189*x^2*y(x)-126)/(5*x^2*y(x)-5))-7/9*ln(-1+x)-7/9*ln(x^2+x+1)+7/3*ln
(x)-_C1 = 0