4.22.9 \(2 y'(x)^3+x y'(x)-2 y(x)=0\)

ODE
\[ 2 y'(x)^3+x y'(x)-2 y(x)=0 \] ODE Classification

[[_1st_order, _with_linear_symmetries], _dAlembert]

Book solution method
Clairaut’s equation and related types, d’Alembert’s equation (also call Lagrange’s)

Mathematica ✓
cpu = 405.949 (sec), leaf count = 2936

\[\left \{\text {Solve}\left [c_1=\int _1^x\frac {\left (12 K[1]^2+6^{2/3} \left (\sqrt {6} \sqrt {K[1]^3+54 y(x)^2}-18 y(x)\right )^{2/3} K[1]+6^{5/6} \sqrt {K[1]^3+54 y(x)^2} \sqrt [3]{\sqrt {6} \sqrt {K[1]^3+54 y(x)^2}-18 y(x)}\right ) K[1]^2+9 y(x) \left (\sqrt [3]{6} \sqrt [3]{\sqrt {6} \sqrt {K[1]^3+54 y(x)^2}-18 y(x)} K[1]^2+\sqrt [6]{6} \sqrt {K[1]^3+54 y(x)^2} \left (\sqrt {6} \sqrt {K[1]^3+54 y(x)^2}-18 y(x)\right )^{2/3}\right )+27\ 6^{2/3} y(x)^2 \left (\sqrt {6} \sqrt {K[1]^3+54 y(x)^2}-18 y(x)\right )^{2/3}}{6 K[1]^5-81 K[1]^2 y(x)^2}dK[1]+\int _1^{y(x)}\frac {2 \left (6 x^4+6^{2/3} \left (\sqrt {6} \sqrt {x^3+54 K[2]^2}-18 K[2]\right )^{2/3} x^3+\sqrt [3]{6} \sqrt [3]{\sqrt {6} \sqrt {x^3+54 K[2]^2}-18 K[2]} \left (9 K[2]+\sqrt {6} \sqrt {x^3+54 K[2]^2}\right ) x^2+81 K[2]^2 x+9 \sqrt [6]{6} K[2] \left (3 \sqrt {6} K[2]+\sqrt {x^3+54 K[2]^2}\right ) \left (\sqrt {6} \sqrt {x^3+54 K[2]^2}-18 K[2]\right )^{2/3}\right )+9 \left (2 x^4 K[2]-27 x K[2]^3\right ) \int _1^x\frac {2\ 6^{5/6} K[1]^6-2 \sqrt [6]{6} \left (\sqrt {6} \sqrt {K[1]^3+54 K[2]^2}-144 K[2]\right ) \sqrt [3]{\sqrt {6} \sqrt {K[1]^3+54 K[2]^2}-18 K[2]} K[1]^4-9 \sqrt [3]{6} K[2] \left (69 \sqrt {6} K[2]-28 \sqrt {K[1]^3+54 K[2]^2}\right ) K[1]^3+216 K[2] \sqrt {K[1]^3+54 K[2]^2} \left (\sqrt {6} \sqrt {K[1]^3+54 K[2]^2}-18 K[2]\right )^{2/3} K[1]^2-27 \sqrt [6]{6} K[2]^2 \sqrt [3]{\sqrt {6} \sqrt {K[1]^3+54 K[2]^2}-18 K[2]} \left (5 \sqrt {6} \sqrt {K[1]^3+54 K[2]^2}-126 K[2]\right ) K[1]+972 \sqrt [3]{6} K[2]^3 \left (\sqrt {K[1]^3+54 K[2]^2}-3 \sqrt {6} K[2]\right )}{\left (2 K[1]^3-27 K[2]^2\right )^2 \sqrt {K[1]^3+54 K[2]^2} \left (\sqrt {6} \sqrt {K[1]^3+54 K[2]^2}-18 K[2]\right )^{2/3}}dK[1]}{9 \left (27 x K[2]^3-2 x^4 K[2]\right )}dK[2],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {\left (24 K[3]^2-\sqrt [6]{3} \left (3 i+\sqrt {3}\right ) \left (2 \sqrt {6} \sqrt {K[3]^3+54 y(x)^2}-36 y(x)\right )^{2/3} K[3]-2^{5/6} \left (-3 i+\sqrt {3}\right ) \sqrt {K[3]^3+54 y(x)^2} \sqrt [3]{3 \sqrt {6} \sqrt {K[3]^3+54 y(x)^2}-54 y(x)}\right ) K[3]^2+9 i y(x) \left (\sqrt [3]{6} \left (i+\sqrt {3}\right ) K[3]^2 \sqrt [3]{\sqrt {6} \sqrt {K[3]^3+54 y(x)^2}-18 y(x)}-\sqrt [6]{6} \left (-i+\sqrt {3}\right ) \sqrt {K[3]^3+54 y(x)^2} \left (\sqrt {6} \sqrt {K[3]^3+54 y(x)^2}-18 y(x)\right )^{2/3}\right )-27 \sqrt [6]{3} \left (3 i+\sqrt {3}\right ) y(x)^2 \left (2 \sqrt {6} \sqrt {K[3]^3+54 y(x)^2}-36 y(x)\right )^{2/3}}{6 K[3]^2 \left (2 K[3]^3-27 y(x)^2\right )}dK[3]+\int _1^{y(x)}\frac {-12 x^4+\sqrt [6]{3} \left (3 i+\sqrt {3}\right ) \left (2 \sqrt {6} \sqrt {x^3+54 K[4]^2}-36 K[4]\right )^{2/3} x^3+\sqrt [3]{6} \sqrt [3]{\sqrt {6} \sqrt {x^3+54 K[4]^2}-18 K[4]} \left (\left (9-9 i \sqrt {3}\right ) K[4]+\sqrt {2} \left (-3 i+\sqrt {3}\right ) \sqrt {x^3+54 K[4]^2}\right ) x^2-162 K[4]^2 x+9 \sqrt [6]{6} K[4] \left (\sqrt {6} \sqrt {x^3+54 K[4]^2}-18 K[4]\right )^{2/3} \left (3 \sqrt {2} \left (3 i+\sqrt {3}\right ) K[4]+\left (1+i \sqrt {3}\right ) \sqrt {x^3+54 K[4]^2}\right )-9 \left (2 x^4 K[4]-27 x K[4]^3\right ) \int _1^x\frac {-2 2^{5/6} \sqrt [3]{3} \left (-3 i+\sqrt {3}\right ) K[3]^6+2 \sqrt [6]{6} \sqrt [3]{\sqrt {6} \sqrt {K[3]^3+54 K[4]^2}-18 K[4]} \left (\sqrt {2} \left (3 i+\sqrt {3}\right ) \sqrt {K[3]^3+54 K[4]^2}-144 i \left (-i+\sqrt {3}\right ) K[4]\right ) K[3]^4+9 \sqrt [3]{6} K[4] \left (69 \sqrt {2} \left (-3 i+\sqrt {3}\right ) K[4]+28 i \left (i+\sqrt {3}\right ) \sqrt {K[3]^3+54 K[4]^2}\right ) K[3]^3+432 K[4] \sqrt {K[3]^3+54 K[4]^2} \left (\sqrt {6} \sqrt {K[3]^3+54 K[4]^2}-18 K[4]\right )^{2/3} K[3]^2+27 \sqrt [6]{6} K[4]^2 \sqrt [3]{\sqrt {6} \sqrt {K[3]^3+54 K[4]^2}-18 K[4]} \left (5 \sqrt {2} \left (3 i+\sqrt {3}\right ) \sqrt {K[3]^3+54 K[4]^2}-126 i \left (-i+\sqrt {3}\right ) K[4]\right ) K[3]+972 \sqrt [3]{6} K[4]^3 \left (3 \sqrt {2} \left (-3 i+\sqrt {3}\right ) K[4]+i \left (i+\sqrt {3}\right ) \sqrt {K[3]^3+54 K[4]^2}\right )}{2 \left (2 K[3]^3-27 K[4]^2\right )^2 \sqrt {K[3]^3+54 K[4]^2} \left (\sqrt {6} \sqrt {K[3]^3+54 K[4]^2}-18 K[4]\right )^{2/3}}dK[3]}{9 \left (2 x^4 K[4]-27 x K[4]^3\right )}dK[4],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {\left (24 K[5]^2-\sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) \left (2 \sqrt {6} \sqrt {K[5]^3+54 y(x)^2}-36 y(x)\right )^{2/3} K[5]-2^{5/6} \left (3 i+\sqrt {3}\right ) \sqrt {K[5]^3+54 y(x)^2} \sqrt [3]{3 \sqrt {6} \sqrt {K[5]^3+54 y(x)^2}-54 y(x)}\right ) K[5]^2+9 i \sqrt [6]{6} y(x) \sqrt [3]{\sqrt {6} \sqrt {K[5]^3+54 y(x)^2}-18 y(x)} \left (\left (i+\sqrt {3}\right ) \sqrt {K[5]^3+54 y(x)^2} \sqrt [3]{\sqrt {6} \sqrt {K[5]^3+54 y(x)^2}-18 y(x)}-\sqrt [6]{6} \left (-i+\sqrt {3}\right ) K[5]^2\right )-27 \sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) y(x)^2 \left (2 \sqrt {6} \sqrt {K[5]^3+54 y(x)^2}-36 y(x)\right )^{2/3}}{6 K[5]^2 \left (2 K[5]^3-27 y(x)^2\right )}dK[5]+\int _1^{y(x)}\frac {-12 x^4+\sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) \left (2 \sqrt {6} \sqrt {x^3+54 K[6]^2}-36 K[6]\right )^{2/3} x^3+\sqrt [3]{6} \sqrt [3]{\sqrt {6} \sqrt {x^3+54 K[6]^2}-18 K[6]} \left (\left (9+9 i \sqrt {3}\right ) K[6]+\sqrt {2} \left (3 i+\sqrt {3}\right ) \sqrt {x^3+54 K[6]^2}\right ) x^2-162 K[6]^2 x+9 \sqrt [6]{6} K[6] \left (\sqrt {6} \sqrt {x^3+54 K[6]^2}-18 K[6]\right )^{2/3} \left (3 \sqrt {2} \left (-3 i+\sqrt {3}\right ) K[6]+\left (1-i \sqrt {3}\right ) \sqrt {x^3+54 K[6]^2}\right )-9 \left (2 x^4 K[6]-27 x K[6]^3\right ) \int _1^x\frac {-2 2^{5/6} \sqrt [3]{3} \left (3 i+\sqrt {3}\right ) K[5]^6+2 \sqrt [6]{6} \sqrt [3]{\sqrt {6} \sqrt {K[5]^3+54 K[6]^2}-18 K[6]} \left (144 i \left (i+\sqrt {3}\right ) K[6]+\sqrt {2} \left (-3 i+\sqrt {3}\right ) \sqrt {K[5]^3+54 K[6]^2}\right ) K[5]^4+9 \sqrt [3]{6} K[6] \left (69 \sqrt {2} \left (3 i+\sqrt {3}\right ) K[6]-28 i \left (-i+\sqrt {3}\right ) \sqrt {K[5]^3+54 K[6]^2}\right ) K[5]^3+432 K[6] \sqrt {K[5]^3+54 K[6]^2} \left (\sqrt {6} \sqrt {K[5]^3+54 K[6]^2}-18 K[6]\right )^{2/3} K[5]^2+27 \sqrt [6]{6} K[6]^2 \sqrt [3]{\sqrt {6} \sqrt {K[5]^3+54 K[6]^2}-18 K[6]} \left (126 i \left (i+\sqrt {3}\right ) K[6]+5 \sqrt {2} \left (-3 i+\sqrt {3}\right ) \sqrt {K[5]^3+54 K[6]^2}\right ) K[5]+972 \sqrt [3]{6} K[6]^3 \left (3 \sqrt {2} \left (3 i+\sqrt {3}\right ) K[6]+\left (-1-i \sqrt {3}\right ) \sqrt {K[5]^3+54 K[6]^2}\right )}{2 \left (2 K[5]^3-27 K[6]^2\right )^2 \sqrt {K[5]^3+54 K[6]^2} \left (\sqrt {6} \sqrt {K[5]^3+54 K[6]^2}-18 K[6]\right )^{2/3}}dK[5]}{9 \left (2 x^4 K[6]-27 x K[6]^3\right )}dK[6],y(x)\right ]\right \}\]

Maple ✓
cpu = 0.057 (sec), leaf count = 79

\[\left [y \left (x \right ) = \left (-\frac {\textit {\_C1}}{12}-\frac {\sqrt {\textit {\_C1}^{2}+24 x}}{12}\right )^{3}+\frac {x \left (-\frac {\textit {\_C1}}{12}-\frac {\sqrt {\textit {\_C1}^{2}+24 x}}{12}\right )}{2}, y \left (x \right ) = \left (-\frac {\textit {\_C1}}{12}+\frac {\sqrt {\textit {\_C1}^{2}+24 x}}{12}\right )^{3}+\frac {x \left (-\frac {\textit {\_C1}}{12}+\frac {\sqrt {\textit {\_C1}^{2}+24 x}}{12}\right )}{2}\right ]\] Mathematica raw input

DSolve[-2*y[x] + x*y'[x] + 2*y'[x]^3 == 0,y[x],x]

Mathematica raw output

{Solve[C[1] == Inactive[Integrate][(27*6^(2/3)*y[x]^2*(-18*y[x] + Sqrt[6]*Sqrt[K
[1]^3 + 54*y[x]^2])^(2/3) + K[1]^2*(12*K[1]^2 + 6^(5/6)*Sqrt[K[1]^3 + 54*y[x]^2]
*(-18*y[x] + Sqrt[6]*Sqrt[K[1]^3 + 54*y[x]^2])^(1/3) + 6^(2/3)*K[1]*(-18*y[x] + 
Sqrt[6]*Sqrt[K[1]^3 + 54*y[x]^2])^(2/3)) + 9*y[x]*(6^(1/3)*K[1]^2*(-18*y[x] + Sq
rt[6]*Sqrt[K[1]^3 + 54*y[x]^2])^(1/3) + 6^(1/6)*Sqrt[K[1]^3 + 54*y[x]^2]*(-18*y[
x] + Sqrt[6]*Sqrt[K[1]^3 + 54*y[x]^2])^(2/3)))/(6*K[1]^5 - 81*K[1]^2*y[x]^2), {K
[1], 1, x}] + Inactive[Integrate][(2*(6*x^4 + 81*x*K[2]^2 + 6^(2/3)*x^3*(-18*K[2
] + Sqrt[6]*Sqrt[x^3 + 54*K[2]^2])^(2/3) + 9*6^(1/6)*K[2]*(3*Sqrt[6]*K[2] + Sqrt
[x^3 + 54*K[2]^2])*(-18*K[2] + Sqrt[6]*Sqrt[x^3 + 54*K[2]^2])^(2/3) + 6^(1/3)*x^
2*(-18*K[2] + Sqrt[6]*Sqrt[x^3 + 54*K[2]^2])^(1/3)*(9*K[2] + Sqrt[6]*Sqrt[x^3 + 
54*K[2]^2])) + 9*(2*x^4*K[2] - 27*x*K[2]^3)*Inactive[Integrate][(2*6^(5/6)*K[1]^
6 - 9*6^(1/3)*K[1]^3*K[2]*(69*Sqrt[6]*K[2] - 28*Sqrt[K[1]^3 + 54*K[2]^2]) + 972*
6^(1/3)*K[2]^3*(-3*Sqrt[6]*K[2] + Sqrt[K[1]^3 + 54*K[2]^2]) - 2*6^(1/6)*K[1]^4*(
-144*K[2] + Sqrt[6]*Sqrt[K[1]^3 + 54*K[2]^2])*(-18*K[2] + Sqrt[6]*Sqrt[K[1]^3 + 
54*K[2]^2])^(1/3) + 216*K[1]^2*K[2]*Sqrt[K[1]^3 + 54*K[2]^2]*(-18*K[2] + Sqrt[6]
*Sqrt[K[1]^3 + 54*K[2]^2])^(2/3) - 27*6^(1/6)*K[1]*K[2]^2*(-18*K[2] + Sqrt[6]*Sq
rt[K[1]^3 + 54*K[2]^2])^(1/3)*(-126*K[2] + 5*Sqrt[6]*Sqrt[K[1]^3 + 54*K[2]^2]))/
((2*K[1]^3 - 27*K[2]^2)^2*Sqrt[K[1]^3 + 54*K[2]^2]*(-18*K[2] + Sqrt[6]*Sqrt[K[1]
^3 + 54*K[2]^2])^(2/3)), {K[1], 1, x}])/(9*(-2*x^4*K[2] + 27*x*K[2]^3)), {K[2], 
1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(-27*3^(1/6)*(3*I + Sqrt[3])
*y[x]^2*(-36*y[x] + 2*Sqrt[6]*Sqrt[K[3]^3 + 54*y[x]^2])^(2/3) + (9*I)*y[x]*(6^(1
/3)*(I + Sqrt[3])*K[3]^2*(-18*y[x] + Sqrt[6]*Sqrt[K[3]^3 + 54*y[x]^2])^(1/3) - 6
^(1/6)*(-I + Sqrt[3])*Sqrt[K[3]^3 + 54*y[x]^2]*(-18*y[x] + Sqrt[6]*Sqrt[K[3]^3 +
 54*y[x]^2])^(2/3)) + K[3]^2*(24*K[3]^2 - 3^(1/6)*(3*I + Sqrt[3])*K[3]*(-36*y[x]
 + 2*Sqrt[6]*Sqrt[K[3]^3 + 54*y[x]^2])^(2/3) - 2^(5/6)*(-3*I + Sqrt[3])*Sqrt[K[3
]^3 + 54*y[x]^2]*(-54*y[x] + 3*Sqrt[6]*Sqrt[K[3]^3 + 54*y[x]^2])^(1/3)))/(6*K[3]
^2*(2*K[3]^3 - 27*y[x]^2)), {K[3], 1, x}] + Inactive[Integrate][(-12*x^4 - 162*x
*K[4]^2 + 3^(1/6)*(3*I + Sqrt[3])*x^3*(-36*K[4] + 2*Sqrt[6]*Sqrt[x^3 + 54*K[4]^2
])^(2/3) + 9*6^(1/6)*K[4]*(-18*K[4] + Sqrt[6]*Sqrt[x^3 + 54*K[4]^2])^(2/3)*(3*Sq
rt[2]*(3*I + Sqrt[3])*K[4] + (1 + I*Sqrt[3])*Sqrt[x^3 + 54*K[4]^2]) + 6^(1/3)*x^
2*(-18*K[4] + Sqrt[6]*Sqrt[x^3 + 54*K[4]^2])^(1/3)*((9 - (9*I)*Sqrt[3])*K[4] + S
qrt[2]*(-3*I + Sqrt[3])*Sqrt[x^3 + 54*K[4]^2]) - 9*(2*x^4*K[4] - 27*x*K[4]^3)*In
active[Integrate][(-2*2^(5/6)*3^(1/3)*(-3*I + Sqrt[3])*K[3]^6 + 432*K[3]^2*K[4]*
Sqrt[K[3]^3 + 54*K[4]^2]*(-18*K[4] + Sqrt[6]*Sqrt[K[3]^3 + 54*K[4]^2])^(2/3) + 9
72*6^(1/3)*K[4]^3*(3*Sqrt[2]*(-3*I + Sqrt[3])*K[4] + I*(I + Sqrt[3])*Sqrt[K[3]^3
 + 54*K[4]^2]) + 9*6^(1/3)*K[3]^3*K[4]*(69*Sqrt[2]*(-3*I + Sqrt[3])*K[4] + (28*I
)*(I + Sqrt[3])*Sqrt[K[3]^3 + 54*K[4]^2]) + 2*6^(1/6)*K[3]^4*(-18*K[4] + Sqrt[6]
*Sqrt[K[3]^3 + 54*K[4]^2])^(1/3)*((-144*I)*(-I + Sqrt[3])*K[4] + Sqrt[2]*(3*I + 
Sqrt[3])*Sqrt[K[3]^3 + 54*K[4]^2]) + 27*6^(1/6)*K[3]*K[4]^2*(-18*K[4] + Sqrt[6]*
Sqrt[K[3]^3 + 54*K[4]^2])^(1/3)*((-126*I)*(-I + Sqrt[3])*K[4] + 5*Sqrt[2]*(3*I +
 Sqrt[3])*Sqrt[K[3]^3 + 54*K[4]^2]))/(2*(2*K[3]^3 - 27*K[4]^2)^2*Sqrt[K[3]^3 + 5
4*K[4]^2]*(-18*K[4] + Sqrt[6]*Sqrt[K[3]^3 + 54*K[4]^2])^(2/3)), {K[3], 1, x}])/(
9*(2*x^4*K[4] - 27*x*K[4]^3)), {K[4], 1, y[x]}], y[x]], Solve[C[1] == Inactive[I
ntegrate][(-27*3^(1/6)*(-3*I + Sqrt[3])*y[x]^2*(-36*y[x] + 2*Sqrt[6]*Sqrt[K[5]^3
 + 54*y[x]^2])^(2/3) + (9*I)*6^(1/6)*y[x]*(-18*y[x] + Sqrt[6]*Sqrt[K[5]^3 + 54*y
[x]^2])^(1/3)*(-(6^(1/6)*(-I + Sqrt[3])*K[5]^2) + (I + Sqrt[3])*Sqrt[K[5]^3 + 54
*y[x]^2]*(-18*y[x] + Sqrt[6]*Sqrt[K[5]^3 + 54*y[x]^2])^(1/3)) + K[5]^2*(24*K[5]^
2 - 3^(1/6)*(-3*I + Sqrt[3])*K[5]*(-36*y[x] + 2*Sqrt[6]*Sqrt[K[5]^3 + 54*y[x]^2]
)^(2/3) - 2^(5/6)*(3*I + Sqrt[3])*Sqrt[K[5]^3 + 54*y[x]^2]*(-54*y[x] + 3*Sqrt[6]
*Sqrt[K[5]^3 + 54*y[x]^2])^(1/3)))/(6*K[5]^2*(2*K[5]^3 - 27*y[x]^2)), {K[5], 1, 
x}] + Inactive[Integrate][(-12*x^4 - 162*x*K[6]^2 + 3^(1/6)*(-3*I + Sqrt[3])*x^3
*(-36*K[6] + 2*Sqrt[6]*Sqrt[x^3 + 54*K[6]^2])^(2/3) + 9*6^(1/6)*K[6]*(-18*K[6] +
 Sqrt[6]*Sqrt[x^3 + 54*K[6]^2])^(2/3)*(3*Sqrt[2]*(-3*I + Sqrt[3])*K[6] + (1 - I*
Sqrt[3])*Sqrt[x^3 + 54*K[6]^2]) + 6^(1/3)*x^2*(-18*K[6] + Sqrt[6]*Sqrt[x^3 + 54*
K[6]^2])^(1/3)*((9 + (9*I)*Sqrt[3])*K[6] + Sqrt[2]*(3*I + Sqrt[3])*Sqrt[x^3 + 54
*K[6]^2]) - 9*(2*x^4*K[6] - 27*x*K[6]^3)*Inactive[Integrate][(-2*2^(5/6)*3^(1/3)
*(3*I + Sqrt[3])*K[5]^6 + 432*K[5]^2*K[6]*Sqrt[K[5]^3 + 54*K[6]^2]*(-18*K[6] + S
qrt[6]*Sqrt[K[5]^3 + 54*K[6]^2])^(2/3) + 972*6^(1/3)*K[6]^3*(3*Sqrt[2]*(3*I + Sq
rt[3])*K[6] + (-1 - I*Sqrt[3])*Sqrt[K[5]^3 + 54*K[6]^2]) + 9*6^(1/3)*K[5]^3*K[6]
*(69*Sqrt[2]*(3*I + Sqrt[3])*K[6] - (28*I)*(-I + Sqrt[3])*Sqrt[K[5]^3 + 54*K[6]^
2]) + 2*6^(1/6)*K[5]^4*(-18*K[6] + Sqrt[6]*Sqrt[K[5]^3 + 54*K[6]^2])^(1/3)*((144
*I)*(I + Sqrt[3])*K[6] + Sqrt[2]*(-3*I + Sqrt[3])*Sqrt[K[5]^3 + 54*K[6]^2]) + 27
*6^(1/6)*K[5]*K[6]^2*(-18*K[6] + Sqrt[6]*Sqrt[K[5]^3 + 54*K[6]^2])^(1/3)*((126*I
)*(I + Sqrt[3])*K[6] + 5*Sqrt[2]*(-3*I + Sqrt[3])*Sqrt[K[5]^3 + 54*K[6]^2]))/(2*
(2*K[5]^3 - 27*K[6]^2)^2*Sqrt[K[5]^3 + 54*K[6]^2]*(-18*K[6] + Sqrt[6]*Sqrt[K[5]^
3 + 54*K[6]^2])^(2/3)), {K[5], 1, x}])/(9*(2*x^4*K[6] - 27*x*K[6]^3)), {K[6], 1,
 y[x]}], y[x]]}

Maple raw input

dsolve(2*diff(y(x),x)^3+x*diff(y(x),x)-2*y(x) = 0, y(x))

Maple raw output

[y(x) = (-1/12*_C1-1/12*(_C1^2+24*x)^(1/2))^3+1/2*x*(-1/12*_C1-1/12*(_C1^2+24*x)
^(1/2)), y(x) = (-1/12*_C1+1/12*(_C1^2+24*x)^(1/2))^3+1/2*x*(-1/12*_C1+1/12*(_C1
^2+24*x)^(1/2))]