4.15.7 \(y(x) \left (1-2 x^3 y(x)^2\right )+x \left (1-2 x^2 y(x)^3\right ) y'(x)=0\)

ODE
\[ y(x) \left (1-2 x^3 y(x)^2\right )+x \left (1-2 x^2 y(x)^3\right ) y'(x)=0 \] ODE Classification

[_rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.031638 (sec), leaf count = 672

\[\left \{\left \{y(x)\to \frac {c_1 x^2+\frac {x^4 \left (c_1-2 x\right ){}^2}{\sqrt [3]{12 c_1 x^8-6 c_1^2 x^7+c_1^3 x^6+3 \sqrt {3} \sqrt {x^8 \left (-24 c_1 x^4+12 c_1^2 x^3-2 c_1^3 x^2+16 x^5+27\right )}-8 x^9-27 x^4}}+\sqrt [3]{12 c_1 x^8-6 c_1^2 x^7+c_1^3 x^6+3 \sqrt {3} \sqrt {x^8 \left (-24 c_1 x^4+12 c_1^2 x^3-2 c_1^3 x^2+16 x^5+27\right )}-8 x^9-27 x^4}-2 x^3}{6 x^2}\right \},\left \{y(x)\to \frac {2 x^2 \left (c_1-2 x\right )-\frac {i \left (\sqrt {3}-i\right ) x^4 \left (c_1-2 x\right ){}^2}{\sqrt [3]{12 c_1 x^8-6 c_1^2 x^7+c_1^3 x^6+3 \sqrt {3} \sqrt {x^8 \left (-24 c_1 x^4+12 c_1^2 x^3-2 c_1^3 x^2+16 x^5+27\right )}-8 x^9-27 x^4}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{12 c_1 x^8-6 c_1^2 x^7+c_1^3 x^6+3 \sqrt {3} \sqrt {x^8 \left (-24 c_1 x^4+12 c_1^2 x^3-2 c_1^3 x^2+16 x^5+27\right )}-8 x^9-27 x^4}}{12 x^2}\right \},\left \{y(x)\to \frac {2 x^2 \left (c_1-2 x\right )+\frac {i \left (\sqrt {3}+i\right ) x^4 \left (c_1-2 x\right ){}^2}{\sqrt [3]{12 c_1 x^8-6 c_1^2 x^7+c_1^3 x^6+3 \sqrt {3} \sqrt {x^8 \left (-24 c_1 x^4+12 c_1^2 x^3-2 c_1^3 x^2+16 x^5+27\right )}-8 x^9-27 x^4}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{12 c_1 x^8-6 c_1^2 x^7+c_1^3 x^6+3 \sqrt {3} \sqrt {x^8 \left (-24 c_1 x^4+12 c_1^2 x^3-2 c_1^3 x^2+16 x^5+27\right )}-8 x^9-27 x^4}}{12 x^2}\right \}\right \}\]

Maple
cpu = 0.119 (sec), leaf count = 21

\[ \left \{ -2\,x-{\frac {1}{2\,{x}^{2} \left (y \relax (x ) \right ) ^{2}}}-2\,y \relax (x ) +{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[y[x]*(1 - 2*x^3*y[x]^2) + x*(1 - 2*x^2*y[x]^3)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (-2*x^3 + x^2*C[1] + (x^4*(-2*x + C[1])^2)/(-27*x^4 - 8*x^9 + 12*x^8*C
[1] - 6*x^7*C[1]^2 + x^6*C[1]^3 + 3*Sqrt[3]*Sqrt[x^8*(27 + 16*x^5 - 24*x^4*C[1] 
+ 12*x^3*C[1]^2 - 2*x^2*C[1]^3)])^(1/3) + (-27*x^4 - 8*x^9 + 12*x^8*C[1] - 6*x^7
*C[1]^2 + x^6*C[1]^3 + 3*Sqrt[3]*Sqrt[x^8*(27 + 16*x^5 - 24*x^4*C[1] + 12*x^3*C[
1]^2 - 2*x^2*C[1]^3)])^(1/3))/(6*x^2)}, {y[x] -> (2*x^2*(-2*x + C[1]) - (I*(-I +
 Sqrt[3])*x^4*(-2*x + C[1])^2)/(-27*x^4 - 8*x^9 + 12*x^8*C[1] - 6*x^7*C[1]^2 + x
^6*C[1]^3 + 3*Sqrt[3]*Sqrt[x^8*(27 + 16*x^5 - 24*x^4*C[1] + 12*x^3*C[1]^2 - 2*x^
2*C[1]^3)])^(1/3) + I*(I + Sqrt[3])*(-27*x^4 - 8*x^9 + 12*x^8*C[1] - 6*x^7*C[1]^
2 + x^6*C[1]^3 + 3*Sqrt[3]*Sqrt[x^8*(27 + 16*x^5 - 24*x^4*C[1] + 12*x^3*C[1]^2 -
 2*x^2*C[1]^3)])^(1/3))/(12*x^2)}, {y[x] -> (2*x^2*(-2*x + C[1]) + (I*(I + Sqrt[
3])*x^4*(-2*x + C[1])^2)/(-27*x^4 - 8*x^9 + 12*x^8*C[1] - 6*x^7*C[1]^2 + x^6*C[1
]^3 + 3*Sqrt[3]*Sqrt[x^8*(27 + 16*x^5 - 24*x^4*C[1] + 12*x^3*C[1]^2 - 2*x^2*C[1]
^3)])^(1/3) - (1 + I*Sqrt[3])*(-27*x^4 - 8*x^9 + 12*x^8*C[1] - 6*x^7*C[1]^2 + x^
6*C[1]^3 + 3*Sqrt[3]*Sqrt[x^8*(27 + 16*x^5 - 24*x^4*C[1] + 12*x^3*C[1]^2 - 2*x^2
*C[1]^3)])^(1/3))/(12*x^2)}}

Maple raw input

dsolve(x*(1-2*x^2*y(x)^3)*diff(y(x),x)+(1-2*x^3*y(x)^2)*y(x) = 0, y(x),'implicit')

Maple raw output

-2*x-1/2/y(x)^2/x^2-2*y(x)+_C1 = 0