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

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

[_exact, _rational]

Book solution method
Exact equation

Mathematica
cpu = 0.115507 (sec), leaf count = 1807

\[\left \{\left \{y(x)\to -\frac {\sqrt {-2 x^2+\sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}+\frac {4 \left (x^4-3 c_1\right )}{\sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}}}}{\sqrt {6}}-\frac {1}{2} \sqrt {-\frac {8 x^2}{3}-\frac {4 \sqrt {6} x}{\sqrt {-2 x^2+\sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}+\frac {4 \left (x^4-3 c_1\right )}{\sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}}}}-\frac {2}{3} \sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}-\frac {8 \left (x^4-3 c_1\right )}{3 \sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}}}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {-\frac {8 x^2}{3}-\frac {4 \sqrt {6} x}{\sqrt {-2 x^2+\sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}+\frac {4 \left (x^4-3 c_1\right )}{\sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}}}}-\frac {2}{3} \sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}-\frac {8 \left (x^4-3 c_1\right )}{3 \sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}}}-\frac {\sqrt {-2 x^2+\sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}+\frac {4 \left (x^4-3 c_1\right )}{\sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}}}}{\sqrt {6}}\right \},\left \{y(x)\to \frac {\sqrt {-2 x^2+\sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}+\frac {4 \left (x^4-3 c_1\right )}{\sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}}}}{\sqrt {6}}-\frac {1}{2} \sqrt {-\frac {8 x^2}{3}+\frac {4 \sqrt {6} x}{\sqrt {-2 x^2+\sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}+\frac {4 \left (x^4-3 c_1\right )}{\sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}}}}-\frac {2}{3} \sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}-\frac {8 \left (x^4-3 c_1\right )}{3 \sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}}}\right \},\left \{y(x)\to \frac {\sqrt {-2 x^2+\sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}+\frac {4 \left (x^4-3 c_1\right )}{\sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}}}}{\sqrt {6}}+\frac {1}{2} \sqrt {-\frac {8 x^2}{3}+\frac {4 \sqrt {6} x}{\sqrt {-2 x^2+\sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}+\frac {4 \left (x^4-3 c_1\right )}{\sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}}}}-\frac {2}{3} \sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}-\frac {8 \left (x^4-3 c_1\right )}{3 \sqrt [3]{-8 x^6+9 \left (4 c_1+3\right ) x^2+3 \sqrt {3} \sqrt {-16 x^8+\left (-16 c_1^2+72 c_1+27\right ) x^4+64 c_1^3}}}}\right \}\right \}\]

Maple
cpu = 0.016 (sec), leaf count = 29

\[ \left \{ -{\frac {{x}^{4}}{4}}-{\frac {{x}^{2} \left (y \relax (x ) \right ) ^{2}}{2}}+xy \relax (x ) -{\frac { \left (y \relax (x ) \right ) ^{4}}{4}}+{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> -(Sqrt[-2*x^2 + (4*(x^4 - 3*C[1]))/(-8*x^6 + 9*x^2*(3 + 4*C[1]) + 3*Sq
rt[3]*Sqrt[-16*x^8 + 64*C[1]^3 + x^4*(27 + 72*C[1] - 16*C[1]^2)])^(1/3) + (-8*x^
6 + 9*x^2*(3 + 4*C[1]) + 3*Sqrt[3]*Sqrt[-16*x^8 + 64*C[1]^3 + x^4*(27 + 72*C[1] 
- 16*C[1]^2)])^(1/3)]/Sqrt[6]) - Sqrt[(-8*x^2)/3 - (8*(x^4 - 3*C[1]))/(3*(-8*x^6
 + 9*x^2*(3 + 4*C[1]) + 3*Sqrt[3]*Sqrt[-16*x^8 + 64*C[1]^3 + x^4*(27 + 72*C[1] -
 16*C[1]^2)])^(1/3)) - (2*(-8*x^6 + 9*x^2*(3 + 4*C[1]) + 3*Sqrt[3]*Sqrt[-16*x^8 
+ 64*C[1]^3 + x^4*(27 + 72*C[1] - 16*C[1]^2)])^(1/3))/3 - (4*Sqrt[6]*x)/Sqrt[-2*
x^2 + (4*(x^4 - 3*C[1]))/(-8*x^6 + 9*x^2*(3 + 4*C[1]) + 3*Sqrt[3]*Sqrt[-16*x^8 +
 64*C[1]^3 + x^4*(27 + 72*C[1] - 16*C[1]^2)])^(1/3) + (-8*x^6 + 9*x^2*(3 + 4*C[1
]) + 3*Sqrt[3]*Sqrt[-16*x^8 + 64*C[1]^3 + x^4*(27 + 72*C[1] - 16*C[1]^2)])^(1/3)
]]/2}, {y[x] -> -(Sqrt[-2*x^2 + (4*(x^4 - 3*C[1]))/(-8*x^6 + 9*x^2*(3 + 4*C[1]) 
+ 3*Sqrt[3]*Sqrt[-16*x^8 + 64*C[1]^3 + x^4*(27 + 72*C[1] - 16*C[1]^2)])^(1/3) + 
(-8*x^6 + 9*x^2*(3 + 4*C[1]) + 3*Sqrt[3]*Sqrt[-16*x^8 + 64*C[1]^3 + x^4*(27 + 72
*C[1] - 16*C[1]^2)])^(1/3)]/Sqrt[6]) + Sqrt[(-8*x^2)/3 - (8*(x^4 - 3*C[1]))/(3*(
-8*x^6 + 9*x^2*(3 + 4*C[1]) + 3*Sqrt[3]*Sqrt[-16*x^8 + 64*C[1]^3 + x^4*(27 + 72*
C[1] - 16*C[1]^2)])^(1/3)) - (2*(-8*x^6 + 9*x^2*(3 + 4*C[1]) + 3*Sqrt[3]*Sqrt[-1
6*x^8 + 64*C[1]^3 + x^4*(27 + 72*C[1] - 16*C[1]^2)])^(1/3))/3 - (4*Sqrt[6]*x)/Sq
rt[-2*x^2 + (4*(x^4 - 3*C[1]))/(-8*x^6 + 9*x^2*(3 + 4*C[1]) + 3*Sqrt[3]*Sqrt[-16
*x^8 + 64*C[1]^3 + x^4*(27 + 72*C[1] - 16*C[1]^2)])^(1/3) + (-8*x^6 + 9*x^2*(3 +
 4*C[1]) + 3*Sqrt[3]*Sqrt[-16*x^8 + 64*C[1]^3 + x^4*(27 + 72*C[1] - 16*C[1]^2)])
^(1/3)]]/2}, {y[x] -> Sqrt[-2*x^2 + (4*(x^4 - 3*C[1]))/(-8*x^6 + 9*x^2*(3 + 4*C[
1]) + 3*Sqrt[3]*Sqrt[-16*x^8 + 64*C[1]^3 + x^4*(27 + 72*C[1] - 16*C[1]^2)])^(1/3
) + (-8*x^6 + 9*x^2*(3 + 4*C[1]) + 3*Sqrt[3]*Sqrt[-16*x^8 + 64*C[1]^3 + x^4*(27 
+ 72*C[1] - 16*C[1]^2)])^(1/3)]/Sqrt[6] - Sqrt[(-8*x^2)/3 - (8*(x^4 - 3*C[1]))/(
3*(-8*x^6 + 9*x^2*(3 + 4*C[1]) + 3*Sqrt[3]*Sqrt[-16*x^8 + 64*C[1]^3 + x^4*(27 + 
72*C[1] - 16*C[1]^2)])^(1/3)) - (2*(-8*x^6 + 9*x^2*(3 + 4*C[1]) + 3*Sqrt[3]*Sqrt
[-16*x^8 + 64*C[1]^3 + x^4*(27 + 72*C[1] - 16*C[1]^2)])^(1/3))/3 + (4*Sqrt[6]*x)
/Sqrt[-2*x^2 + (4*(x^4 - 3*C[1]))/(-8*x^6 + 9*x^2*(3 + 4*C[1]) + 3*Sqrt[3]*Sqrt[
-16*x^8 + 64*C[1]^3 + x^4*(27 + 72*C[1] - 16*C[1]^2)])^(1/3) + (-8*x^6 + 9*x^2*(
3 + 4*C[1]) + 3*Sqrt[3]*Sqrt[-16*x^8 + 64*C[1]^3 + x^4*(27 + 72*C[1] - 16*C[1]^2
)])^(1/3)]]/2}, {y[x] -> Sqrt[-2*x^2 + (4*(x^4 - 3*C[1]))/(-8*x^6 + 9*x^2*(3 + 4
*C[1]) + 3*Sqrt[3]*Sqrt[-16*x^8 + 64*C[1]^3 + x^4*(27 + 72*C[1] - 16*C[1]^2)])^(
1/3) + (-8*x^6 + 9*x^2*(3 + 4*C[1]) + 3*Sqrt[3]*Sqrt[-16*x^8 + 64*C[1]^3 + x^4*(
27 + 72*C[1] - 16*C[1]^2)])^(1/3)]/Sqrt[6] + Sqrt[(-8*x^2)/3 - (8*(x^4 - 3*C[1])
)/(3*(-8*x^6 + 9*x^2*(3 + 4*C[1]) + 3*Sqrt[3]*Sqrt[-16*x^8 + 64*C[1]^3 + x^4*(27
 + 72*C[1] - 16*C[1]^2)])^(1/3)) - (2*(-8*x^6 + 9*x^2*(3 + 4*C[1]) + 3*Sqrt[3]*S
qrt[-16*x^8 + 64*C[1]^3 + x^4*(27 + 72*C[1] - 16*C[1]^2)])^(1/3))/3 + (4*Sqrt[6]
*x)/Sqrt[-2*x^2 + (4*(x^4 - 3*C[1]))/(-8*x^6 + 9*x^2*(3 + 4*C[1]) + 3*Sqrt[3]*Sq
rt[-16*x^8 + 64*C[1]^3 + x^4*(27 + 72*C[1] - 16*C[1]^2)])^(1/3) + (-8*x^6 + 9*x^
2*(3 + 4*C[1]) + 3*Sqrt[3]*Sqrt[-16*x^8 + 64*C[1]^3 + x^4*(27 + 72*C[1] - 16*C[1
]^2)])^(1/3)]]/2}}

Maple raw input

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

Maple raw output

-1/4*x^4-1/2*x^2*y(x)^2+x*y(x)-1/4*y(x)^4+_C1 = 0