4.14.40 \(2 x^3+3 x^2 y(x)+\left (-x^3+x^2+3 x y(x)^2+2 y(x)^3\right ) y'(x)-y(x)^3+y(x)^2=0\)

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

[_rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.0442185 (sec), leaf count = 364

\[\left \{\left \{y(x)\to \frac {6 \sqrt [3]{2} \left (c_1+x\right )-2^{2/3} \left (\sqrt {729 \left (c_1 x+x^3\right ){}^2+108 \left (c_1+x\right ){}^3}+27 c_1 x+27 x^3\right ){}^{2/3}}{6 \sqrt [3]{\sqrt {729 \left (c_1 x+x^3\right ){}^2+108 \left (c_1+x\right ){}^3}+27 c_1 x+27 x^3}}\right \},\left \{y(x)\to \frac {2^{2/3} \left (1-i \sqrt {3}\right ) \left (\sqrt {729 \left (c_1 x+x^3\right ){}^2+108 \left (c_1+x\right ){}^3}+27 c_1 x+27 x^3\right ){}^{2/3}-6 i \sqrt [3]{2} \left (\sqrt {3}-i\right ) \left (c_1+x\right )}{12 \sqrt [3]{\sqrt {729 \left (c_1 x+x^3\right ){}^2+108 \left (c_1+x\right ){}^3}+27 c_1 x+27 x^3}}\right \},\left \{y(x)\to \frac {2^{2/3} \left (1+i \sqrt {3}\right ) \left (\sqrt {729 \left (c_1 x+x^3\right ){}^2+108 \left (c_1+x\right ){}^3}+27 c_1 x+27 x^3\right ){}^{2/3}+6 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (c_1+x\right )}{12 \sqrt [3]{\sqrt {729 \left (c_1 x+x^3\right ){}^2+108 \left (c_1+x\right ){}^3}+27 c_1 x+27 x^3}}\right \}\right \}\]

Maple
cpu = 0.162 (sec), leaf count = 28

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

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

Mathematica raw output

{{y[x] -> (6*2^(1/3)*(x + C[1]) - 2^(2/3)*(27*x^3 + 27*x*C[1] + Sqrt[108*(x + C[
1])^3 + 729*(x^3 + x*C[1])^2])^(2/3))/(6*(27*x^3 + 27*x*C[1] + Sqrt[108*(x + C[1
])^3 + 729*(x^3 + x*C[1])^2])^(1/3))}, {y[x] -> ((-6*I)*2^(1/3)*(-I + Sqrt[3])*(
x + C[1]) + 2^(2/3)*(1 - I*Sqrt[3])*(27*x^3 + 27*x*C[1] + Sqrt[108*(x + C[1])^3 
+ 729*(x^3 + x*C[1])^2])^(2/3))/(12*(27*x^3 + 27*x*C[1] + Sqrt[108*(x + C[1])^3 
+ 729*(x^3 + x*C[1])^2])^(1/3))}, {y[x] -> ((6*I)*2^(1/3)*(I + Sqrt[3])*(x + C[1
]) + 2^(2/3)*(1 + I*Sqrt[3])*(27*x^3 + 27*x*C[1] + Sqrt[108*(x + C[1])^3 + 729*(
x^3 + x*C[1])^2])^(2/3))/(12*(27*x^3 + 27*x*C[1] + Sqrt[108*(x + C[1])^3 + 729*(
x^3 + x*C[1])^2])^(1/3))}}

Maple raw input

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

Maple raw output

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