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

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

[_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

Book solution method
Exact equation

Mathematica
cpu = 0.0197385 (sec), leaf count = 414

\[\left \{\left \{y(x)\to \frac {1}{18} \left (-\frac {18 \sqrt [3]{2} \left (x^3+1\right )}{\sqrt [3]{\sqrt {-18 c_1 x^3+3 c_1 \left (3 c_1-4\right )-4 x^9-3 x^6}+3 c_1-3 x^3-2}}-9\ 2^{2/3} \sqrt [3]{\sqrt {-18 c_1 x^3+3 c_1 \left (3 c_1-4\right )-4 x^9-3 x^6}+3 c_1-3 x^3-2}+18\right )\right \},\left \{y(x)\to \frac {1}{36} \left (\frac {18 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) \left (x^3+1\right )}{\sqrt [3]{\sqrt {-18 c_1 x^3+3 c_1 \left (3 c_1-4\right )-4 x^9-3 x^6}+3 c_1-3 x^3-2}}+9\ 2^{2/3} \left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {-18 c_1 x^3+3 c_1 \left (3 c_1-4\right )-4 x^9-3 x^6}+3 c_1-3 x^3-2}+36\right )\right \},\left \{y(x)\to \frac {1}{36} \left (\frac {18 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) \left (x^3+1\right )}{\sqrt [3]{\sqrt {-18 c_1 x^3+3 c_1 \left (3 c_1-4\right )-4 x^9-3 x^6}+3 c_1-3 x^3-2}}+9\ 2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {-18 c_1 x^3+3 c_1 \left (3 c_1-4\right )-4 x^9-3 x^6}+3 c_1-3 x^3-2}+36\right )\right \}\right \}\]

Maple
cpu = 0.013 (sec), leaf count = 21

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

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

Mathematica raw output

{{y[x] -> (18 - (18*2^(1/3)*(1 + x^3))/(-2 - 3*x^3 + 3*C[1] + Sqrt[-3*x^6 - 4*x^
9 - 18*x^3*C[1] + 3*C[1]*(-4 + 3*C[1])])^(1/3) - 9*2^(2/3)*(-2 - 3*x^3 + 3*C[1] 
+ Sqrt[-3*x^6 - 4*x^9 - 18*x^3*C[1] + 3*C[1]*(-4 + 3*C[1])])^(1/3))/18}, {y[x] -
> (36 + (18*2^(1/3)*(1 + I*Sqrt[3])*(1 + x^3))/(-2 - 3*x^3 + 3*C[1] + Sqrt[-3*x^
6 - 4*x^9 - 18*x^3*C[1] + 3*C[1]*(-4 + 3*C[1])])^(1/3) + 9*2^(2/3)*(1 - I*Sqrt[3
])*(-2 - 3*x^3 + 3*C[1] + Sqrt[-3*x^6 - 4*x^9 - 18*x^3*C[1] + 3*C[1]*(-4 + 3*C[1
])])^(1/3))/36}, {y[x] -> (36 + (18*2^(1/3)*(1 - I*Sqrt[3])*(1 + x^3))/(-2 - 3*x
^3 + 3*C[1] + Sqrt[-3*x^6 - 4*x^9 - 18*x^3*C[1] + 3*C[1]*(-4 + 3*C[1])])^(1/3) +
 9*2^(2/3)*(1 + I*Sqrt[3])*(-2 - 3*x^3 + 3*C[1] + Sqrt[-3*x^6 - 4*x^9 - 18*x^3*C
[1] + 3*C[1]*(-4 + 3*C[1])])^(1/3))/36}}

Maple raw input

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

Maple raw output

x^3*y(x)-1/3*y(x)^3+y(x)^2+_C1 = 0