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 = 0.449108 (sec), leaf count = 1807
\[\left \{\left \{y(x)\to -\frac {\sqrt {-2 x^2+\sqrt [3]{-8 x^6+9 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 (4 c_1+3) 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 = 0.027 (sec), leaf count = 29
\[\left [-\frac {x^{4}}{4}-\frac {x^{2} y \left (x \right )^{2}}{2}+x y \left (x \right )-\frac {y \left (x \right )^{4}}{4}+\textit {\_C1} = 0\right ]\] 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))
Maple raw output
[-1/4*x^4-1/2*x^2*y(x)^2+x*y(x)-1/4*y(x)^4+_C1 = 0]