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

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

[_dAlembert]

Book solution method
No Missing Variables ODE, Solve for \(y\)

Mathematica ✓
cpu = 42.1306 (sec), leaf count = 1484

\[\left \{\left \{y(x)\to \frac {-16 x^4+8 \left (\sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}-12\right ) x^3-4 \left (\left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27\right ){}^{2/3}-9 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+54\right ) x^2+6 \left (72 c_1+2 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27\right ){}^{2/3}+4 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+9 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}\right ) x+3 \left (4 c_1 \left (2 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27\right ){}^{2/3}+9 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+54\right )+9 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27\right ){}^{2/3}+12 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+2 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )} \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+27 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+81\right )}{24 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27\right ){}^{2/3}}\right \},\left \{y(x)\to -\frac {2 x^2}{3}+\frac {1}{6} i \left (i+\sqrt {3}\right ) \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27} x-\frac {i \left (-i+\sqrt {3}\right ) (2 x+3)^2 x}{6 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}}+\frac {1}{96} \left (-\frac {i \left (-i+\sqrt {3}\right ) (2 x+3)^2}{\sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}}-4 x+i \left (i+\sqrt {3}\right ) \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+6\right ){}^2+c_1\right \},\left \{y(x)\to -\frac {2 x^2}{3}-\frac {1}{6} i \left (-i+\sqrt {3}\right ) \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27} x+\frac {i \left (i+\sqrt {3}\right ) (2 x+3)^2 x}{6 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}}+\frac {1}{96} \left (\frac {i \left (i+\sqrt {3}\right ) (2 x+3)^2}{\sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}}-4 x-i \left (-i+\sqrt {3}\right ) \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+6\right ){}^2+c_1\right \}\right \}\]

Maple ✓
cpu = 0.138 (sec), leaf count = 1473

\[\left [y \left (x \right ) = 0, y \left (x \right ) = \left (\frac {\left (-36 x^{2}-54 x +108 \textit {\_C1} -8 x^{3}+27+6 \sqrt {-48 \textit {\_C1} \,x^{3}-216 x^{2} \textit {\_C1} -24 x^{3}+324 \textit {\_C1}^{2}-324 x \textit {\_C1} -108 x^{2}+162 \textit {\_C1} -162 x}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (-\frac {1}{3} x -\frac {1}{9} x^{2}-\frac {1}{4}\right )}{\left (-36 x^{2}-54 x +108 \textit {\_C1} -8 x^{3}+27+6 \sqrt {-48 \textit {\_C1} \,x^{3}-216 x^{2} \textit {\_C1} -24 x^{3}+324 \textit {\_C1}^{2}-324 x \textit {\_C1} -108 x^{2}+162 \textit {\_C1} -162 x}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {1}{2}\right )^{3}+x \left (\frac {\left (-36 x^{2}-54 x +108 \textit {\_C1} -8 x^{3}+27+6 \sqrt {-48 \textit {\_C1} \,x^{3}-216 x^{2} \textit {\_C1} -24 x^{3}+324 \textit {\_C1}^{2}-324 x \textit {\_C1} -108 x^{2}+162 \textit {\_C1} -162 x}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (-\frac {1}{3} x -\frac {1}{9} x^{2}-\frac {1}{4}\right )}{\left (-36 x^{2}-54 x +108 \textit {\_C1} -8 x^{3}+27+6 \sqrt {-48 \textit {\_C1} \,x^{3}-216 x^{2} \textit {\_C1} -24 x^{3}+324 \textit {\_C1}^{2}-324 x \textit {\_C1} -108 x^{2}+162 \textit {\_C1} -162 x}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {1}{2}\right )^{2}, y \left (x \right ) = \left (-\frac {\left (-36 x^{2}-54 x +108 \textit {\_C1} -8 x^{3}+27+6 \sqrt {-48 \textit {\_C1} \,x^{3}-216 x^{2} \textit {\_C1} -24 x^{3}+324 \textit {\_C1}^{2}-324 x \textit {\_C1} -108 x^{2}+162 \textit {\_C1} -162 x}\right )^{\frac {1}{3}}}{12}+\frac {-x -\frac {1}{3} x^{2}-\frac {3}{4}}{\left (-36 x^{2}-54 x +108 \textit {\_C1} -8 x^{3}+27+6 \sqrt {-48 \textit {\_C1} \,x^{3}-216 x^{2} \textit {\_C1} -24 x^{3}+324 \textit {\_C1}^{2}-324 x \textit {\_C1} -108 x^{2}+162 \textit {\_C1} -162 x}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {1}{2}-\frac {i \sqrt {3}\, \left (\frac {\left (-36 x^{2}-54 x +108 \textit {\_C1} -8 x^{3}+27+6 \sqrt {-48 \textit {\_C1} \,x^{3}-216 x^{2} \textit {\_C1} -24 x^{3}+324 \textit {\_C1}^{2}-324 x \textit {\_C1} -108 x^{2}+162 \textit {\_C1} -162 x}\right )^{\frac {1}{3}}}{6}+\frac {-2 x -\frac {2}{3} x^{2}-\frac {3}{2}}{\left (-36 x^{2}-54 x +108 \textit {\_C1} -8 x^{3}+27+6 \sqrt {-48 \textit {\_C1} \,x^{3}-216 x^{2} \textit {\_C1} -24 x^{3}+324 \textit {\_C1}^{2}-324 x \textit {\_C1} -108 x^{2}+162 \textit {\_C1} -162 x}\right )^{\frac {1}{3}}}\right )}{2}\right )^{3}+x \left (-\frac {\left (-36 x^{2}-54 x +108 \textit {\_C1} -8 x^{3}+27+6 \sqrt {-48 \textit {\_C1} \,x^{3}-216 x^{2} \textit {\_C1} -24 x^{3}+324 \textit {\_C1}^{2}-324 x \textit {\_C1} -108 x^{2}+162 \textit {\_C1} -162 x}\right )^{\frac {1}{3}}}{12}+\frac {-x -\frac {1}{3} x^{2}-\frac {3}{4}}{\left (-36 x^{2}-54 x +108 \textit {\_C1} -8 x^{3}+27+6 \sqrt {-48 \textit {\_C1} \,x^{3}-216 x^{2} \textit {\_C1} -24 x^{3}+324 \textit {\_C1}^{2}-324 x \textit {\_C1} -108 x^{2}+162 \textit {\_C1} -162 x}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {1}{2}-\frac {i \sqrt {3}\, \left (\frac {\left (-36 x^{2}-54 x +108 \textit {\_C1} -8 x^{3}+27+6 \sqrt {-48 \textit {\_C1} \,x^{3}-216 x^{2} \textit {\_C1} -24 x^{3}+324 \textit {\_C1}^{2}-324 x \textit {\_C1} -108 x^{2}+162 \textit {\_C1} -162 x}\right )^{\frac {1}{3}}}{6}+\frac {-2 x -\frac {2}{3} x^{2}-\frac {3}{2}}{\left (-36 x^{2}-54 x +108 \textit {\_C1} -8 x^{3}+27+6 \sqrt {-48 \textit {\_C1} \,x^{3}-216 x^{2} \textit {\_C1} -24 x^{3}+324 \textit {\_C1}^{2}-324 x \textit {\_C1} -108 x^{2}+162 \textit {\_C1} -162 x}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}, y \left (x \right ) = \left (-\frac {\left (-36 x^{2}-54 x +108 \textit {\_C1} -8 x^{3}+27+6 \sqrt {-48 \textit {\_C1} \,x^{3}-216 x^{2} \textit {\_C1} -24 x^{3}+324 \textit {\_C1}^{2}-324 x \textit {\_C1} -108 x^{2}+162 \textit {\_C1} -162 x}\right )^{\frac {1}{3}}}{12}+\frac {-x -\frac {1}{3} x^{2}-\frac {3}{4}}{\left (-36 x^{2}-54 x +108 \textit {\_C1} -8 x^{3}+27+6 \sqrt {-48 \textit {\_C1} \,x^{3}-216 x^{2} \textit {\_C1} -24 x^{3}+324 \textit {\_C1}^{2}-324 x \textit {\_C1} -108 x^{2}+162 \textit {\_C1} -162 x}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {1}{2}+\frac {i \sqrt {3}\, \left (\frac {\left (-36 x^{2}-54 x +108 \textit {\_C1} -8 x^{3}+27+6 \sqrt {-48 \textit {\_C1} \,x^{3}-216 x^{2} \textit {\_C1} -24 x^{3}+324 \textit {\_C1}^{2}-324 x \textit {\_C1} -108 x^{2}+162 \textit {\_C1} -162 x}\right )^{\frac {1}{3}}}{6}+\frac {-2 x -\frac {2}{3} x^{2}-\frac {3}{2}}{\left (-36 x^{2}-54 x +108 \textit {\_C1} -8 x^{3}+27+6 \sqrt {-48 \textit {\_C1} \,x^{3}-216 x^{2} \textit {\_C1} -24 x^{3}+324 \textit {\_C1}^{2}-324 x \textit {\_C1} -108 x^{2}+162 \textit {\_C1} -162 x}\right )^{\frac {1}{3}}}\right )}{2}\right )^{3}+x \left (-\frac {\left (-36 x^{2}-54 x +108 \textit {\_C1} -8 x^{3}+27+6 \sqrt {-48 \textit {\_C1} \,x^{3}-216 x^{2} \textit {\_C1} -24 x^{3}+324 \textit {\_C1}^{2}-324 x \textit {\_C1} -108 x^{2}+162 \textit {\_C1} -162 x}\right )^{\frac {1}{3}}}{12}+\frac {-x -\frac {1}{3} x^{2}-\frac {3}{4}}{\left (-36 x^{2}-54 x +108 \textit {\_C1} -8 x^{3}+27+6 \sqrt {-48 \textit {\_C1} \,x^{3}-216 x^{2} \textit {\_C1} -24 x^{3}+324 \textit {\_C1}^{2}-324 x \textit {\_C1} -108 x^{2}+162 \textit {\_C1} -162 x}\right )^{\frac {1}{3}}}-\frac {x}{3}+\frac {1}{2}+\frac {i \sqrt {3}\, \left (\frac {\left (-36 x^{2}-54 x +108 \textit {\_C1} -8 x^{3}+27+6 \sqrt {-48 \textit {\_C1} \,x^{3}-216 x^{2} \textit {\_C1} -24 x^{3}+324 \textit {\_C1}^{2}-324 x \textit {\_C1} -108 x^{2}+162 \textit {\_C1} -162 x}\right )^{\frac {1}{3}}}{6}+\frac {-2 x -\frac {2}{3} x^{2}-\frac {3}{2}}{\left (-36 x^{2}-54 x +108 \textit {\_C1} -8 x^{3}+27+6 \sqrt {-48 \textit {\_C1} \,x^{3}-216 x^{2} \textit {\_C1} -24 x^{3}+324 \textit {\_C1}^{2}-324 x \textit {\_C1} -108 x^{2}+162 \textit {\_C1} -162 x}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ]\] Mathematica raw input

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

Mathematica raw output

{{y[x] -> (-16*x^4 + 8*x^3*(-12 + (27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqr
t[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3)) - 4*x^2*(54 -
 9*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x -
 18*x^2 - 4*x^3 + 27*C[1])])^(1/3) + (27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*
Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(2/3)) + 6*x*(72*
C[1] + 4*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])] + 9*(27 -
 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 
- 4*x^3 + 27*C[1])])^(1/3) + 2*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6
]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(2/3)) + 3*(81 + 12*Sqr
t[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])] + 27*(27 - 54*x - 36*
x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 2
7*C[1])])^(1/3) + 2*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])
]*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 
18*x^2 - 4*x^3 + 27*C[1])])^(1/3) + 9*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6
*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(2/3) + 4*C[1]*(
54 + 9*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27
*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3) + 2*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1
] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(2/3))))/(2
4*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 
18*x^2 - 4*x^3 + 27*C[1])])^(2/3))}, {y[x] -> (-2*x^2)/3 + C[1] - ((I/6)*(-I + S
qrt[3])*x*(3 + 2*x)^2)/(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(
1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3) + (I/6)*(I + Sqrt[3])*x*(
27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*
x^2 - 4*x^3 + 27*C[1])])^(1/3) + (6 - 4*x - (I*(-I + Sqrt[3])*(3 + 2*x)^2)/(27 -
 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 
- 4*x^3 + 27*C[1])])^(1/3) + I*(I + Sqrt[3])*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C
[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3))^2/
96}, {y[x] -> (-2*x^2)/3 + C[1] + ((I/6)*(I + Sqrt[3])*x*(3 + 2*x)^2)/(27 - 54*x
 - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x
^3 + 27*C[1])])^(1/3) - (I/6)*(-I + Sqrt[3])*x*(27 - 54*x - 36*x^2 - 8*x^3 + 108
*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3) +
 (6 - 4*x + (I*(I + Sqrt[3])*(3 + 2*x)^2)/(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1]
 + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3) - I*(-
I + Sqrt[3])*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1]
)*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3))^2/96}}

Maple raw input

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

Maple raw output

[y(x) = 0, y(x) = (1/6*(-36*x^2-54*x+108*_C1-8*x^3+27+6*(-48*_C1*x^3-216*_C1*x^2
-24*x^3+324*_C1^2-324*_C1*x-108*x^2+162*_C1-162*x)^(1/2))^(1/3)-6*(-1/3*x-1/9*x^
2-1/4)/(-36*x^2-54*x+108*_C1-8*x^3+27+6*(-48*_C1*x^3-216*_C1*x^2-24*x^3+324*_C1^
2-324*_C1*x-108*x^2+162*_C1-162*x)^(1/2))^(1/3)-1/3*x+1/2)^3+x*(1/6*(-36*x^2-54*
x+108*_C1-8*x^3+27+6*(-48*_C1*x^3-216*_C1*x^2-24*x^3+324*_C1^2-324*_C1*x-108*x^2
+162*_C1-162*x)^(1/2))^(1/3)-6*(-1/3*x-1/9*x^2-1/4)/(-36*x^2-54*x+108*_C1-8*x^3+
27+6*(-48*_C1*x^3-216*_C1*x^2-24*x^3+324*_C1^2-324*_C1*x-108*x^2+162*_C1-162*x)^
(1/2))^(1/3)-1/3*x+1/2)^2, y(x) = (-1/12*(-36*x^2-54*x+108*_C1-8*x^3+27+6*(-48*_
C1*x^3-216*_C1*x^2-24*x^3+324*_C1^2-324*_C1*x-108*x^2+162*_C1-162*x)^(1/2))^(1/3
)+3*(-1/3*x-1/9*x^2-1/4)/(-36*x^2-54*x+108*_C1-8*x^3+27+6*(-48*_C1*x^3-216*_C1*x
^2-24*x^3+324*_C1^2-324*_C1*x-108*x^2+162*_C1-162*x)^(1/2))^(1/3)-1/3*x+1/2-1/2*
I*3^(1/2)*(1/6*(-36*x^2-54*x+108*_C1-8*x^3+27+6*(-48*_C1*x^3-216*_C1*x^2-24*x^3+
324*_C1^2-324*_C1*x-108*x^2+162*_C1-162*x)^(1/2))^(1/3)+6*(-1/3*x-1/9*x^2-1/4)/(
-36*x^2-54*x+108*_C1-8*x^3+27+6*(-48*_C1*x^3-216*_C1*x^2-24*x^3+324*_C1^2-324*_C
1*x-108*x^2+162*_C1-162*x)^(1/2))^(1/3)))^3+x*(-1/12*(-36*x^2-54*x+108*_C1-8*x^3
+27+6*(-48*_C1*x^3-216*_C1*x^2-24*x^3+324*_C1^2-324*_C1*x-108*x^2+162*_C1-162*x)
^(1/2))^(1/3)+3*(-1/3*x-1/9*x^2-1/4)/(-36*x^2-54*x+108*_C1-8*x^3+27+6*(-48*_C1*x
^3-216*_C1*x^2-24*x^3+324*_C1^2-324*_C1*x-108*x^2+162*_C1-162*x)^(1/2))^(1/3)-1/
3*x+1/2-1/2*I*3^(1/2)*(1/6*(-36*x^2-54*x+108*_C1-8*x^3+27+6*(-48*_C1*x^3-216*_C1
*x^2-24*x^3+324*_C1^2-324*_C1*x-108*x^2+162*_C1-162*x)^(1/2))^(1/3)+6*(-1/3*x-1/
9*x^2-1/4)/(-36*x^2-54*x+108*_C1-8*x^3+27+6*(-48*_C1*x^3-216*_C1*x^2-24*x^3+324*
_C1^2-324*_C1*x-108*x^2+162*_C1-162*x)^(1/2))^(1/3)))^2, y(x) = (-1/12*(-36*x^2-
54*x+108*_C1-8*x^3+27+6*(-48*_C1*x^3-216*_C1*x^2-24*x^3+324*_C1^2-324*_C1*x-108*
x^2+162*_C1-162*x)^(1/2))^(1/3)+3*(-1/3*x-1/9*x^2-1/4)/(-36*x^2-54*x+108*_C1-8*x
^3+27+6*(-48*_C1*x^3-216*_C1*x^2-24*x^3+324*_C1^2-324*_C1*x-108*x^2+162*_C1-162*
x)^(1/2))^(1/3)-1/3*x+1/2+1/2*I*3^(1/2)*(1/6*(-36*x^2-54*x+108*_C1-8*x^3+27+6*(-
48*_C1*x^3-216*_C1*x^2-24*x^3+324*_C1^2-324*_C1*x-108*x^2+162*_C1-162*x)^(1/2))^
(1/3)+6*(-1/3*x-1/9*x^2-1/4)/(-36*x^2-54*x+108*_C1-8*x^3+27+6*(-48*_C1*x^3-216*_
C1*x^2-24*x^3+324*_C1^2-324*_C1*x-108*x^2+162*_C1-162*x)^(1/2))^(1/3)))^3+x*(-1/
12*(-36*x^2-54*x+108*_C1-8*x^3+27+6*(-48*_C1*x^3-216*_C1*x^2-24*x^3+324*_C1^2-32
4*_C1*x-108*x^2+162*_C1-162*x)^(1/2))^(1/3)+3*(-1/3*x-1/9*x^2-1/4)/(-36*x^2-54*x
+108*_C1-8*x^3+27+6*(-48*_C1*x^3-216*_C1*x^2-24*x^3+324*_C1^2-324*_C1*x-108*x^2+
162*_C1-162*x)^(1/2))^(1/3)-1/3*x+1/2+1/2*I*3^(1/2)*(1/6*(-36*x^2-54*x+108*_C1-8
*x^3+27+6*(-48*_C1*x^3-216*_C1*x^2-24*x^3+324*_C1^2-324*_C1*x-108*x^2+162*_C1-16
2*x)^(1/2))^(1/3)+6*(-1/3*x-1/9*x^2-1/4)/(-36*x^2-54*x+108*_C1-8*x^3+27+6*(-48*_
C1*x^3-216*_C1*x^2-24*x^3+324*_C1^2-324*_C1*x-108*x^2+162*_C1-162*x)^(1/2))^(1/3
)))^2]