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

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

[[_homogeneous, `class G`], _exact, _rational]

Book solution method
Exact equation

Mathematica
cpu = 0.0192639 (sec), leaf count = 328

\[\left \{\left \{y(x)\to -\frac {\sqrt [3]{2} \left (9 c_1 x^2+\sqrt {81 c_1^2 x^4-12 x^9}\right ){}^{2/3}+2 \sqrt [3]{3} x^3}{6^{2/3} x \sqrt [3]{9 c_1 x^2+\sqrt {81 c_1^2 x^4-12 x^9}}}\right \},\left \{y(x)\to \frac {\sqrt [3]{3} \left (1-i \sqrt {3}\right ) \left (18 c_1 x^2+2 \sqrt {81 c_1^2 x^4-12 x^9}\right ){}^{2/3}+2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) x^3}{12 x \sqrt [3]{9 c_1 x^2+\sqrt {81 c_1^2 x^4-12 x^9}}}\right \},\left \{y(x)\to \frac {\sqrt [3]{3} \left (1+i \sqrt {3}\right ) \left (18 c_1 x^2+2 \sqrt {81 c_1^2 x^4-12 x^9}\right ){}^{2/3}+2 \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) x^3}{12 x \sqrt [3]{9 c_1 x^2+\sqrt {81 c_1^2 x^4-12 x^9}}}\right \}\right \}\]

Maple
cpu = 0.014 (sec), leaf count = 27

\[ \left \{ \ln \relax (x ) -{\it \_C1}+{\frac {2}{5}\ln \left (-{y \relax (x ) \left (x- \left (y \relax (x ) \right ) ^{2} \right ) {x}^{-{\frac {3}{2}}}} \right ) }=0 \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> -((2*3^(1/3)*x^3 + 2^(1/3)*(9*x^2*C[1] + Sqrt[-12*x^9 + 81*x^4*C[1]^2]
)^(2/3))/(6^(2/3)*x*(9*x^2*C[1] + Sqrt[-12*x^9 + 81*x^4*C[1]^2])^(1/3)))}, {y[x]
 -> (2*2^(1/3)*3^(1/6)*(3*I + Sqrt[3])*x^3 + 3^(1/3)*(1 - I*Sqrt[3])*(18*x^2*C[1
] + 2*Sqrt[-12*x^9 + 81*x^4*C[1]^2])^(2/3))/(12*x*(9*x^2*C[1] + Sqrt[-12*x^9 + 8
1*x^4*C[1]^2])^(1/3))}, {y[x] -> (2*2^(1/3)*3^(1/6)*(-3*I + Sqrt[3])*x^3 + 3^(1/
3)*(1 + I*Sqrt[3])*(18*x^2*C[1] + 2*Sqrt[-12*x^9 + 81*x^4*C[1]^2])^(2/3))/(12*x*
(9*x^2*C[1] + Sqrt[-12*x^9 + 81*x^4*C[1]^2])^(1/3))}}

Maple raw input

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

Maple raw output

ln(x)-_C1+2/5*ln(-y(x)*(x-y(x)^2)/x^(3/2)) = 0