4.12.36 \(\left (1-x^3 y(x)\right ) y'(x)=x^2 y(x)^2\)

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

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type``class B`]]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 14.689 (sec), leaf count = 326

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

Maple
cpu = 0.019 (sec), leaf count = 30

\[ \left \{ \ln \relax (x ) -{\it \_C1}-{\frac {\ln \left (2\,{x}^{3}y \relax (x ) -3 \right ) }{6}}-{\frac {\ln \left ({x}^{3}y \relax (x ) \right ) }{3}}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> (1 + (1 + 12*x^6*C[1] + 2*Sqrt[6]*Sqrt[x^6*C[1]*(1 + 6*x^6*C[1])])^(-1
/3) + (1 + 12*x^6*C[1] + 2*Sqrt[6]*Sqrt[x^6*C[1]*(1 + 6*x^6*C[1])])^(1/3))/(2*x^
3)}, {y[x] -> (4 - (2*(1 + I*Sqrt[3]))/(1 + 12*x^6*C[1] + 2*Sqrt[6]*Sqrt[x^6*C[1
]*(1 + 6*x^6*C[1])])^(1/3) + (2*I)*(I + Sqrt[3])*(1 + 12*x^6*C[1] + 2*Sqrt[6]*Sq
rt[x^6*C[1]*(1 + 6*x^6*C[1])])^(1/3))/(8*x^3)}, {y[x] -> (4 + ((2*I)*(I + Sqrt[3
]))/(1 + 12*x^6*C[1] + 2*Sqrt[6]*Sqrt[x^6*C[1]*(1 + 6*x^6*C[1])])^(1/3) - 2*(1 +
 I*Sqrt[3])*(1 + 12*x^6*C[1] + 2*Sqrt[6]*Sqrt[x^6*C[1]*(1 + 6*x^6*C[1])])^(1/3))
/(8*x^3)}}

Maple raw input

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

Maple raw output

ln(x)-_C1-1/6*ln(2*x^3*y(x)-3)-1/3*ln(x^3*y(x)) = 0