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