ODE
\[ \left (-2 y(x)^4-x y(x)^3+4 x\right ) y'(x)=y(x) \left (y(x)^3+2\right ) \] ODE Classification
[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
Book solution method
Exact equation, integrating factor
Mathematica ✓
cpu = 0.528858 (sec), leaf count = 2021
\[\left \{\left \{y(x)\to -\frac {x}{4}-\frac {1}{2} \sqrt {\frac {x^2}{4}+\frac {2 c_1}{3}+\frac {\sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (c_1{}^2+24 x\right )}{3 \sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}}-\frac {1}{2} \sqrt {\frac {x^2}{2}+\frac {\left (x^2+4 c_1\right ) x}{4 \sqrt {\frac {x^2}{4}+\frac {2 c_1}{3}+\frac {\sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (c_1{}^2+24 x\right )}{3 \sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}}}+\frac {4 c_1}{3}-\frac {\sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (c_1{}^2+24 x\right )}{3 \sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}}\right \},\left \{y(x)\to -\frac {x}{4}-\frac {1}{2} \sqrt {\frac {x^2}{4}+\frac {2 c_1}{3}+\frac {\sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (c_1{}^2+24 x\right )}{3 \sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}}+\frac {1}{2} \sqrt {\frac {x^2}{2}+\frac {\left (x^2+4 c_1\right ) x}{4 \sqrt {\frac {x^2}{4}+\frac {2 c_1}{3}+\frac {\sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (c_1{}^2+24 x\right )}{3 \sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}}}+\frac {4 c_1}{3}-\frac {\sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (c_1{}^2+24 x\right )}{3 \sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}}\right \},\left \{y(x)\to -\frac {x}{4}+\frac {1}{2} \sqrt {\frac {x^2}{4}+\frac {2 c_1}{3}+\frac {\sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (c_1{}^2+24 x\right )}{3 \sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}}-\frac {1}{2} \sqrt {\frac {x^2}{2}-\frac {\left (x^2+4 c_1\right ) x}{4 \sqrt {\frac {x^2}{4}+\frac {2 c_1}{3}+\frac {\sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (c_1{}^2+24 x\right )}{3 \sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}}}+\frac {4 c_1}{3}-\frac {\sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (c_1{}^2+24 x\right )}{3 \sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}}\right \},\left \{y(x)\to -\frac {x}{4}+\frac {1}{2} \sqrt {\frac {x^2}{4}+\frac {2 c_1}{3}+\frac {\sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (c_1{}^2+24 x\right )}{3 \sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}}+\frac {1}{2} \sqrt {\frac {x^2}{2}-\frac {\left (x^2+4 c_1\right ) x}{4 \sqrt {\frac {x^2}{4}+\frac {2 c_1}{3}+\frac {\sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}{3 \sqrt [3]{2}}+\frac {\sqrt [3]{2} \left (c_1{}^2+24 x\right )}{3 \sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}}}+\frac {4 c_1}{3}-\frac {\sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (c_1{}^2+24 x\right )}{3 \sqrt [3]{54 x^3+144 c_1 x-2 c_1{}^3+\sqrt {\left (54 x^3+144 c_1 x-2 c_1{}^3\right ){}^2-4 \left (c_1{}^2+24 x\right ){}^3}}}}\right \}\right \}\]
Maple ✓
cpu = 0.037 (sec), leaf count = 27
\[\left [x -\frac {\left (-y \left (x \right )^{2}+\textit {\_C1} \right ) y \left (x \right )^{2}}{2+y \left (x \right )^{3}} = 0\right ]\] Mathematica raw input
DSolve[(4*x - x*y[x]^3 - 2*y[x]^4)*y'[x] == y[x]*(2 + y[x]^3),y[x],x]
Mathematica raw output
{{y[x] -> -1/4*x - Sqrt[x^2/4 + (2*C[1])/3 + (2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)) + (54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)/(3*2^(1/3))]/2 - Sqrt[x^2/2 + (4*C[1])/3 - (2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)) - (54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)/(3*2^(1/3)) + (x*(x^2 + 4*C[1]))/(4*Sqrt[x^2/4 + (2*C[1])/3 + (2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)) + (54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)/(3*2^(1/3))])]/2}, {y[x] -> -1/4*x - Sqrt[x^2/4 + (2*C[1])/3 + (2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)) + (54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)/(3*2^(1/3))]/2 + Sqrt[x^2/2 + (4*C[1])/3 - (2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)) - (54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)/(3*2^(1/3)) + (x*(x^2 + 4*C[1]))/(4*Sqrt[x^2/4 + (2*C[1])/3 + (2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)) + (54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)/(3*2^(1/3))])]/2}, {y[x] -> -1/4*x + Sqrt[x^2/4 + (2*C[1])/3 + (2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)) + (54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)/(3*2^(1/3))]/2 - Sqrt[x^2/2 + (4*C[1])/3 - (2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)) - (54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)/(3*2^(1/3)) - (x*(x^2 + 4*C[1]))/(4*Sqrt[x^2/4 + (2*C[1])/3 + (2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)) + (54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)/(3*2^(1/3))])]/2}, {y[x] -> -1/4*x + Sqrt[x^2/4 + (2*C[1])/3 + (2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)) + (54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)/(3*2^(1/3))]/2 + Sqrt[x^2/2 + (4*C[1])/3 - (2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)) - (54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)/(3*2^(1/3)) - (x*(x^2 + 4*C[1]))/(4*Sqrt[x^2/4 + (2*C[1])/3 + (2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)) + (54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)/(3*2^(1/3))])]/2}}
Maple raw input
dsolve((4*x-x*y(x)^3-2*y(x)^4)*diff(y(x),x) = (2+y(x)^3)*y(x), y(x))
Maple raw output
[x-(-y(x)^2+_C1)/(2+y(x)^3)*y(x)^2 = 0]