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

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

[_exact, _rational]

Book solution method
Exact equation

Mathematica
cpu = 0.0951783 (sec), leaf count = 1211

\[\left \{\left \{y(x)\to -\frac {\sqrt {\frac {4 x^3+12 c_1+\left (243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}}}}}{\sqrt {6}}-\frac {1}{2} \sqrt {-\frac {12 \sqrt {6} x}{\sqrt {\frac {4 x^3+12 c_1+\left (243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}}}}}-\frac {2}{3} \sqrt [3]{243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}}-\frac {8 \left (x^3+3 c_1\right )}{3 \sqrt [3]{243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}}}}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {-\frac {12 \sqrt {6} x}{\sqrt {\frac {4 x^3+12 c_1+\left (243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}}}}}-\frac {2}{3} \sqrt [3]{243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}}-\frac {8 \left (x^3+3 c_1\right )}{3 \sqrt [3]{243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}}}}-\frac {\sqrt {\frac {4 x^3+12 c_1+\left (243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}}}}}{\sqrt {6}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {4 x^3+12 c_1+\left (243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}}}}}{\sqrt {6}}-\frac {1}{2} \sqrt {\frac {12 \sqrt {6} x}{\sqrt {\frac {4 x^3+12 c_1+\left (243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}}}}}-\frac {2}{3} \sqrt [3]{243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}}-\frac {8 \left (x^3+3 c_1\right )}{3 \sqrt [3]{243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}}}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {4 x^3+12 c_1+\left (243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}}}}}{\sqrt {6}}+\frac {1}{2} \sqrt {\frac {12 \sqrt {6} x}{\sqrt {\frac {4 x^3+12 c_1+\left (243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}}}}}-\frac {2}{3} \sqrt [3]{243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}}-\frac {8 \left (x^3+3 c_1\right )}{3 \sqrt [3]{243 x^2-\frac {1}{432} \sqrt {11019960576 x^4-4 \left (144 x^3+432 c_1\right ){}^3}}}}\right \}\right \}\]

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

\[ \left \{ -{\frac {{x}^{3}}{3}}+3\,xy \relax (x ) -{\frac { \left (y \relax (x ) \right ) ^{4}}{4}}+{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> -(Sqrt[(4*x^3 + 12*C[1] + (243*x^2 - Sqrt[11019960576*x^4 - 4*(144*x^3
 + 432*C[1])^3]/432)^(2/3))/(243*x^2 - Sqrt[11019960576*x^4 - 4*(144*x^3 + 432*C
[1])^3]/432)^(1/3)]/Sqrt[6]) - Sqrt[(-8*(x^3 + 3*C[1]))/(3*(243*x^2 - Sqrt[11019
960576*x^4 - 4*(144*x^3 + 432*C[1])^3]/432)^(1/3)) - (2*(243*x^2 - Sqrt[11019960
576*x^4 - 4*(144*x^3 + 432*C[1])^3]/432)^(1/3))/3 - (12*Sqrt[6]*x)/Sqrt[(4*x^3 +
 12*C[1] + (243*x^2 - Sqrt[11019960576*x^4 - 4*(144*x^3 + 432*C[1])^3]/432)^(2/3
))/(243*x^2 - Sqrt[11019960576*x^4 - 4*(144*x^3 + 432*C[1])^3]/432)^(1/3)]]/2}, 
{y[x] -> -(Sqrt[(4*x^3 + 12*C[1] + (243*x^2 - Sqrt[11019960576*x^4 - 4*(144*x^3 
+ 432*C[1])^3]/432)^(2/3))/(243*x^2 - Sqrt[11019960576*x^4 - 4*(144*x^3 + 432*C[
1])^3]/432)^(1/3)]/Sqrt[6]) + Sqrt[(-8*(x^3 + 3*C[1]))/(3*(243*x^2 - Sqrt[110199
60576*x^4 - 4*(144*x^3 + 432*C[1])^3]/432)^(1/3)) - (2*(243*x^2 - Sqrt[110199605
76*x^4 - 4*(144*x^3 + 432*C[1])^3]/432)^(1/3))/3 - (12*Sqrt[6]*x)/Sqrt[(4*x^3 + 
12*C[1] + (243*x^2 - Sqrt[11019960576*x^4 - 4*(144*x^3 + 432*C[1])^3]/432)^(2/3)
)/(243*x^2 - Sqrt[11019960576*x^4 - 4*(144*x^3 + 432*C[1])^3]/432)^(1/3)]]/2}, {
y[x] -> Sqrt[(4*x^3 + 12*C[1] + (243*x^2 - Sqrt[11019960576*x^4 - 4*(144*x^3 + 4
32*C[1])^3]/432)^(2/3))/(243*x^2 - Sqrt[11019960576*x^4 - 4*(144*x^3 + 432*C[1])
^3]/432)^(1/3)]/Sqrt[6] - Sqrt[(-8*(x^3 + 3*C[1]))/(3*(243*x^2 - Sqrt[1101996057
6*x^4 - 4*(144*x^3 + 432*C[1])^3]/432)^(1/3)) - (2*(243*x^2 - Sqrt[11019960576*x
^4 - 4*(144*x^3 + 432*C[1])^3]/432)^(1/3))/3 + (12*Sqrt[6]*x)/Sqrt[(4*x^3 + 12*C
[1] + (243*x^2 - Sqrt[11019960576*x^4 - 4*(144*x^3 + 432*C[1])^3]/432)^(2/3))/(2
43*x^2 - Sqrt[11019960576*x^4 - 4*(144*x^3 + 432*C[1])^3]/432)^(1/3)]]/2}, {y[x]
 -> Sqrt[(4*x^3 + 12*C[1] + (243*x^2 - Sqrt[11019960576*x^4 - 4*(144*x^3 + 432*C
[1])^3]/432)^(2/3))/(243*x^2 - Sqrt[11019960576*x^4 - 4*(144*x^3 + 432*C[1])^3]/
432)^(1/3)]/Sqrt[6] + Sqrt[(-8*(x^3 + 3*C[1]))/(3*(243*x^2 - Sqrt[11019960576*x^
4 - 4*(144*x^3 + 432*C[1])^3]/432)^(1/3)) - (2*(243*x^2 - Sqrt[11019960576*x^4 -
 4*(144*x^3 + 432*C[1])^3]/432)^(1/3))/3 + (12*Sqrt[6]*x)/Sqrt[(4*x^3 + 12*C[1] 
+ (243*x^2 - Sqrt[11019960576*x^4 - 4*(144*x^3 + 432*C[1])^3]/432)^(2/3))/(243*x
^2 - Sqrt[11019960576*x^4 - 4*(144*x^3 + 432*C[1])^3]/432)^(1/3)]]/2}}

Maple raw input

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

Maple raw output

-1/3*x^3+3*x*y(x)-1/4*y(x)^4+_C1 = 0