4.15.13 \(\left (-2 y(x)^4-x y(x)^3+4 x\right ) y'(x)=y(x) \left (y(x)^3+2\right )\)

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 = 0.150154 (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 = 0.016 (sec), leaf count = 27

\[ \left \{ x-{\frac { \left (- \left (y \relax (x ) \right ) ^{2}+{\it \_C1} \right ) \left (y \relax (x ) \right ) ^{2}}{2+ \left (y \relax (x ) \right ) ^{3}}}=0 \right \} \] 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] -> -x/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 - 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] -> -x/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 + 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] -> -x/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 + (5
4*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)) + (54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sq
rt[-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 + S
qrt[-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] -> -x/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 + 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),'implicit')

Maple raw output

x-(-y(x)^2+_C1)/(2+y(x)^3)*y(x)^2 = 0