ODE
\[ x^2 y'(x)^3-2 x y(x) y'(x)^2+y(x)^2 y'(x)+1=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries], _Clairaut]
Book solution method
Clairaut’s equation and related types, \(f(y-x y', y')=0\)
Mathematica ✓
cpu = 111.884 (sec), leaf count = 3033
\[\left \{\text {Solve}\left [c_1=\int _1^x\frac {108 y(x)^2 K[1]^6+16 y(x)^5 K[1]^5+4\ 2^{2/3} y(x)^4 \sqrt [3]{-27 K[1]^4-2 y(x)^3 K[1]^3+3 \sqrt {3} \sqrt {K[1]^7 \left (4 y(x)^3+27 K[1]\right )}} K[1]^4-4 \sqrt [3]{2} y(x)^3 \left (-27 K[1]^4-2 y(x)^3 K[1]^3+3 \sqrt {3} \sqrt {K[1]^7 \left (4 y(x)^3+27 K[1]\right )}\right )^{2/3} K[1]^3+3\ 2^{2/3} y(x) \left (9 K[1]^4+\sqrt {3} \sqrt {K[1]^7 \left (4 y(x)^3+27 K[1]\right )}\right ) \sqrt [3]{-27 K[1]^4-2 y(x)^3 K[1]^3+3 \sqrt {3} \sqrt {K[1]^7 \left (4 y(x)^3+27 K[1]\right )}} K[1]-3 \sqrt [3]{2} \left (9 K[1]^4+\sqrt {3} \sqrt {K[1]^7 \left (4 y(x)^3+27 K[1]\right )}\right ) \left (-27 K[1]^4-2 y(x)^3 K[1]^3+3 \sqrt {3} \sqrt {K[1]^7 \left (4 y(x)^3+27 K[1]\right )}\right )^{2/3}}{4 K[1]^6 y(x)^2 \left (4 y(x)^3+27 K[1]\right )}dK[1]+\int _1^{y(x)}\left (-\frac {3 \left (27 x^4+4 K[2]^3 x^3+3 \sqrt {3} \sqrt {x^7 \left (4 K[2]^3+27 x\right )}\right ) \sqrt [3]{-54 x^4-4 K[2]^3 x^3+6 \sqrt {3} \sqrt {x^7 \left (4 K[2]^3+27 x\right )}} \left (\sqrt [3]{2} x K[2]-\sqrt [3]{-27 x^4-2 K[2]^3 x^3+3 \sqrt {3} \sqrt {x^7 \left (4 K[2]^3+27 x\right )}}\right )}{4 x^5 K[2]^3 \left (4 K[2]^3+27 x\right )}-\int _1^x\frac {3 K[1]^8 \left (54 \sqrt [3]{2} \sqrt {3} K[2] K[1]^5+\sqrt {3} \left (2 \sqrt [3]{2} K[2]^4-27 \sqrt [3]{-27 K[1]^4-2 K[2]^3 K[1]^3+3 \sqrt {3} \sqrt {K[1]^7 \left (4 K[2]^3+27 K[1]\right )}}\right ) K[1]^4+2 \sqrt {3} K[2]^3 \sqrt [3]{-27 K[1]^4-2 K[2]^3 K[1]^3+3 \sqrt {3} \sqrt {K[1]^7 \left (4 K[2]^3+27 K[1]\right )}} K[1]^3-18 \sqrt [3]{2} K[2] \sqrt {K[1]^7 \left (4 K[2]^3+27 K[1]\right )} K[1]+9 \sqrt {K[1]^7 \left (4 K[2]^3+27 K[1]\right )} \sqrt [3]{-27 K[1]^4-2 K[2]^3 K[1]^3+3 \sqrt {3} \sqrt {K[1]^7 \left (4 K[2]^3+27 K[1]\right )}}\right )}{\left (K[1]^7 \left (4 K[2]^3+27 K[1]\right )\right )^{3/2} \left (-54 K[1]^4-4 K[2]^3 K[1]^3+6 \sqrt {3} \sqrt {K[1]^7 \left (4 K[2]^3+27 K[1]\right )}\right )^{2/3}}dK[1]\right )dK[2],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {108 \left (i+\sqrt {3}\right ) y(x)^2 K[3]^6+16 \left (i+\sqrt {3}\right ) y(x)^5 K[3]^5-8 i 2^{2/3} y(x)^4 \sqrt [3]{-27 K[3]^4-2 y(x)^3 K[3]^3+3 \sqrt {3} \sqrt {K[3]^7 \left (4 y(x)^3+27 K[3]\right )}} K[3]^4+4 \sqrt [3]{2} \left (-i+\sqrt {3}\right ) y(x)^3 \left (-27 K[3]^4-2 y(x)^3 K[3]^3+3 \sqrt {3} \sqrt {K[3]^7 \left (4 y(x)^3+27 K[3]\right )}\right )^{2/3} K[3]^3-6 i 2^{2/3} y(x) \left (9 K[3]^4+\sqrt {3} \sqrt {K[3]^7 \left (4 y(x)^3+27 K[3]\right )}\right ) \sqrt [3]{-27 K[3]^4-2 y(x)^3 K[3]^3+3 \sqrt {3} \sqrt {K[3]^7 \left (4 y(x)^3+27 K[3]\right )}} K[3]+3 \sqrt [3]{2} \left (-27 K[3]^4-2 y(x)^3 K[3]^3+3 \sqrt {3} \sqrt {K[3]^7 \left (4 y(x)^3+27 K[3]\right )}\right )^{2/3} \left (9 \left (-i+\sqrt {3}\right ) K[3]^4+\left (3-i \sqrt {3}\right ) \sqrt {K[3]^7 \left (4 y(x)^3+27 K[3]\right )}\right )}{4 \left (i+\sqrt {3}\right ) K[3]^6 y(x)^2 \left (4 y(x)^3+27 K[3]\right )}dK[3]+\int _1^{y(x)}\left (\frac {3 i \sqrt [3]{-54 x^4-4 K[4]^3 x^3+6 \sqrt {3} \sqrt {x^7 \left (4 K[4]^3+27 x\right )}} \left (6 \sqrt [3]{2} \sqrt {3} K[4] x^5+3 \left (3 i+\sqrt {3}\right ) \sqrt [3]{-27 x^4-2 K[4]^3 x^3+3 \sqrt {3} \sqrt {x^7 \left (4 K[4]^3+27 x\right )}} x^4+2 \sqrt [3]{2} K[4] \sqrt {x^7 \left (4 K[4]^3+27 x\right )} x+\left (1+i \sqrt {3}\right ) \sqrt {x^7 \left (4 K[4]^3+27 x\right )} \sqrt [3]{-27 x^4-2 K[4]^3 x^3+3 \sqrt {3} \sqrt {x^7 \left (4 K[4]^3+27 x\right )}}\right )}{4 \left (i+\sqrt {3}\right ) x^2 K[4]^3 \sqrt {x^7 \left (4 K[4]^3+27 x\right )}}-\int _1^x\frac {3 K[3]^8 \left (-108 i \sqrt [3]{2} \sqrt {3} K[4] K[3]^5+\left (27 \left (3-i \sqrt {3}\right ) \sqrt [3]{-27 K[3]^4-2 K[4]^3 K[3]^3+3 \sqrt {3} \sqrt {K[3]^7 \left (4 K[4]^3+27 K[3]\right )}}-4 i \sqrt [3]{2} \sqrt {3} K[4]^4\right ) K[3]^4+2 i \left (3 i+\sqrt {3}\right ) K[4]^3 \sqrt [3]{-27 K[3]^4-2 K[4]^3 K[3]^3+3 \sqrt {3} \sqrt {K[3]^7 \left (4 K[4]^3+27 K[3]\right )}} K[3]^3+36 i \sqrt [3]{2} K[4] \sqrt {K[3]^7 \left (4 K[4]^3+27 K[3]\right )} K[3]-9 \left (-i+\sqrt {3}\right ) \sqrt {K[3]^7 \left (4 K[4]^3+27 K[3]\right )} \sqrt [3]{-27 K[3]^4-2 K[4]^3 K[3]^3+3 \sqrt {3} \sqrt {K[3]^7 \left (4 K[4]^3+27 K[3]\right )}}\right )}{\left (i+\sqrt {3}\right ) \left (K[3]^7 \left (4 K[4]^3+27 K[3]\right )\right )^{3/2} \left (-54 K[3]^4-4 K[4]^3 K[3]^3+6 \sqrt {3} \sqrt {K[3]^7 \left (4 K[4]^3+27 K[3]\right )}\right )^{2/3}}dK[3]\right )dK[4],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {108 \left (-i+\sqrt {3}\right ) y(x)^2 K[5]^6+16 \left (-i+\sqrt {3}\right ) y(x)^5 K[5]^5+8 i 2^{2/3} y(x)^4 \sqrt [3]{-27 K[5]^4-2 y(x)^3 K[5]^3+3 \sqrt {3} \sqrt {K[5]^7 \left (4 y(x)^3+27 K[5]\right )}} K[5]^4+4 \sqrt [3]{2} \left (i+\sqrt {3}\right ) y(x)^3 \left (-27 K[5]^4-2 y(x)^3 K[5]^3+3 \sqrt {3} \sqrt {K[5]^7 \left (4 y(x)^3+27 K[5]\right )}\right )^{2/3} K[5]^3+6 i 2^{2/3} y(x) \left (9 K[5]^4+\sqrt {3} \sqrt {K[5]^7 \left (4 y(x)^3+27 K[5]\right )}\right ) \sqrt [3]{-27 K[5]^4-2 y(x)^3 K[5]^3+3 \sqrt {3} \sqrt {K[5]^7 \left (4 y(x)^3+27 K[5]\right )}} K[5]+3 \sqrt [3]{2} \left (-27 K[5]^4-2 y(x)^3 K[5]^3+3 \sqrt {3} \sqrt {K[5]^7 \left (4 y(x)^3+27 K[5]\right )}\right )^{2/3} \left (9 \left (i+\sqrt {3}\right ) K[5]^4+\left (3+i \sqrt {3}\right ) \sqrt {K[5]^7 \left (4 y(x)^3+27 K[5]\right )}\right )}{4 \left (-i+\sqrt {3}\right ) K[5]^6 y(x)^2 \left (4 y(x)^3+27 K[5]\right )}dK[5]+\int _1^{y(x)}\left (-\frac {3 \sqrt [3]{-54 x^4-4 K[6]^3 x^3+6 \sqrt {3} \sqrt {x^7 \left (4 K[6]^3+27 x\right )}} \left (6 i \sqrt [3]{2} \sqrt {3} K[6] x^5+3 \left (3+i \sqrt {3}\right ) \sqrt [3]{-27 x^4-2 K[6]^3 x^3+3 \sqrt {3} \sqrt {x^7 \left (4 K[6]^3+27 x\right )}} x^4+2 i \sqrt [3]{2} K[6] \sqrt {x^7 \left (4 K[6]^3+27 x\right )} x+\left (i+\sqrt {3}\right ) \sqrt {x^7 \left (4 K[6]^3+27 x\right )} \sqrt [3]{-27 x^4-2 K[6]^3 x^3+3 \sqrt {3} \sqrt {x^7 \left (4 K[6]^3+27 x\right )}}\right )}{4 \left (-i+\sqrt {3}\right ) x^2 K[6]^3 \sqrt {x^7 \left (4 K[6]^3+27 x\right )}}-\int _1^x\frac {3 K[5]^8 \left (108 i \sqrt [3]{2} \sqrt {3} K[6] K[5]^5+\left (4 i \sqrt [3]{2} \sqrt {3} K[6]^4+27 \left (3+i \sqrt {3}\right ) \sqrt [3]{-27 K[5]^4-2 K[6]^3 K[5]^3+3 \sqrt {3} \sqrt {K[5]^7 \left (4 K[6]^3+27 K[5]\right )}}\right ) K[5]^4-2 i \left (-3 i+\sqrt {3}\right ) K[6]^3 \sqrt [3]{-27 K[5]^4-2 K[6]^3 K[5]^3+3 \sqrt {3} \sqrt {K[5]^7 \left (4 K[6]^3+27 K[5]\right )}} K[5]^3-36 i \sqrt [3]{2} K[6] \sqrt {K[5]^7 \left (4 K[6]^3+27 K[5]\right )} K[5]-9 \left (i+\sqrt {3}\right ) \sqrt {K[5]^7 \left (4 K[6]^3+27 K[5]\right )} \sqrt [3]{-27 K[5]^4-2 K[6]^3 K[5]^3+3 \sqrt {3} \sqrt {K[5]^7 \left (4 K[6]^3+27 K[5]\right )}}\right )}{\left (-i+\sqrt {3}\right ) \left (K[5]^7 \left (4 K[6]^3+27 K[5]\right )\right )^{3/2} \left (-54 K[5]^4-4 K[6]^3 K[5]^3+6 \sqrt {3} \sqrt {K[5]^7 \left (4 K[6]^3+27 K[5]\right )}\right )^{2/3}}dK[5]\right )dK[6],y(x)\right ]\right \}\]
Maple ✓
cpu = 0.111 (sec), leaf count = 81
\[\left [y \left (x \right ) = \frac {3 \left (-2 x \right )^{\frac {1}{3}}}{2}, y \left (x \right ) = -\frac {3 \left (-2 x \right )^{\frac {1}{3}}}{4}-\frac {3 i \sqrt {3}\, \left (-2 x \right )^{\frac {1}{3}}}{4}, y \left (x \right ) = -\frac {3 \left (-2 x \right )^{\frac {1}{3}}}{4}+\frac {3 i \sqrt {3}\, \left (-2 x \right )^{\frac {1}{3}}}{4}, y \left (x \right ) = x \textit {\_C1} -\frac {1}{\sqrt {-\textit {\_C1}}}, y \left (x \right ) = x \textit {\_C1} +\frac {1}{\sqrt {-\textit {\_C1}}}\right ]\] Mathematica raw input
DSolve[1 + y[x]^2*y'[x] - 2*x*y[x]*y'[x]^2 + x^2*y'[x]^3 == 0,y[x],x]
Mathematica raw output
{Solve[C[1] == Inactive[Integrate][(108*K[1]^6*y[x]^2 + 16*K[1]^5*y[x]^5 + 4*2^(2/3)*K[1]^4*y[x]^4*(-27*K[1]^4 - 2*K[1]^3*y[x]^3 + 3*Sqrt[3]*Sqrt[K[1]^7*(27*K[1] + 4*y[x]^3)])^(1/3) + 3*2^(2/3)*K[1]*y[x]*(9*K[1]^4 + Sqrt[3]*Sqrt[K[1]^7*(27*K[1] + 4*y[x]^3)])*(-27*K[1]^4 - 2*K[1]^3*y[x]^3 + 3*Sqrt[3]*Sqrt[K[1]^7*(27*K[1] + 4*y[x]^3)])^(1/3) - 4*2^(1/3)*K[1]^3*y[x]^3*(-27*K[1]^4 - 2*K[1]^3*y[x]^3 + 3*Sqrt[3]*Sqrt[K[1]^7*(27*K[1] + 4*y[x]^3)])^(2/3) - 3*2^(1/3)*(9*K[1]^4 + Sqrt[3]*Sqrt[K[1]^7*(27*K[1] + 4*y[x]^3)])*(-27*K[1]^4 - 2*K[1]^3*y[x]^3 + 3*Sqrt[3]*Sqrt[K[1]^7*(27*K[1] + 4*y[x]^3)])^(2/3))/(4*K[1]^6*y[x]^2*(27*K[1] + 4*y[x]^3)), {K[1], 1, x}] + Inactive[Integrate][(-3*(27*x^4 + 4*x^3*K[2]^3 + 3*Sqrt[3]*Sqrt[x^7*(27*x + 4*K[2]^3)])*(-54*x^4 - 4*x^3*K[2]^3 + 6*Sqrt[3]*Sqrt[x^7*(27*x + 4*K[2]^3)])^(1/3)*(2^(1/3)*x*K[2] - (-27*x^4 - 2*x^3*K[2]^3 + 3*Sqrt[3]*Sqrt[x^7*(27*x + 4*K[2]^3)])^(1/3)))/(4*x^5*K[2]^3*(27*x + 4*K[2]^3)) - Inactive[Integrate][(3*K[1]^8*(54*2^(1/3)*Sqrt[3]*K[1]^5*K[2] - 18*2^(1/3)*K[1]*K[2]*Sqrt[K[1]^7*(27*K[1] + 4*K[2]^3)] + 2*Sqrt[3]*K[1]^3*K[2]^3*(-27*K[1]^4 - 2*K[1]^3*K[2]^3 + 3*Sqrt[3]*Sqrt[K[1]^7*(27*K[1] + 4*K[2]^3)])^(1/3) + 9*Sqrt[K[1]^7*(27*K[1] + 4*K[2]^3)]*(-27*K[1]^4 - 2*K[1]^3*K[2]^3 + 3*Sqrt[3]*Sqrt[K[1]^7*(27*K[1] + 4*K[2]^3)])^(1/3) + Sqrt[3]*K[1]^4*(2*2^(1/3)*K[2]^4 - 27*(-27*K[1]^4 - 2*K[1]^3*K[2]^3 + 3*Sqrt[3]*Sqrt[K[1]^7*(27*K[1] + 4*K[2]^3)])^(1/3))))/((K[1]^7*(27*K[1] + 4*K[2]^3))^(3/2)*(-54*K[1]^4 - 4*K[1]^3*K[2]^3 + 6*Sqrt[3]*Sqrt[K[1]^7*(27*K[1] + 4*K[2]^3)])^(2/3)), {K[1], 1, x}], {K[2], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(108*(I + Sqrt[3])*K[3]^6*y[x]^2 + 16*(I + Sqrt[3])*K[3]^5*y[x]^5 - (8*I)*2^(2/3)*K[3]^4*y[x]^4*(-27*K[3]^4 - 2*K[3]^3*y[x]^3 + 3*Sqrt[3]*Sqrt[K[3]^7*(27*K[3] + 4*y[x]^3)])^(1/3) - (6*I)*2^(2/3)*K[3]*y[x]*(9*K[3]^4 + Sqrt[3]*Sqrt[K[3]^7*(27*K[3] + 4*y[x]^3)])*(-27*K[3]^4 - 2*K[3]^3*y[x]^3 + 3*Sqrt[3]*Sqrt[K[3]^7*(27*K[3] + 4*y[x]^3)])^(1/3) + 4*2^(1/3)*(-I + Sqrt[3])*K[3]^3*y[x]^3*(-27*K[3]^4 - 2*K[3]^3*y[x]^3 + 3*Sqrt[3]*Sqrt[K[3]^7*(27*K[3] + 4*y[x]^3)])^(2/3) + 3*2^(1/3)*(-27*K[3]^4 - 2*K[3]^3*y[x]^3 + 3*Sqrt[3]*Sqrt[K[3]^7*(27*K[3] + 4*y[x]^3)])^(2/3)*(9*(-I + Sqrt[3])*K[3]^4 + (3 - I*Sqrt[3])*Sqrt[K[3]^7*(27*K[3] + 4*y[x]^3)]))/(4*(I + Sqrt[3])*K[3]^6*y[x]^2*(27*K[3] + 4*y[x]^3)), {K[3], 1, x}] + Inactive[Integrate][(((3*I)/4)*(-54*x^4 - 4*x^3*K[4]^3 + 6*Sqrt[3]*Sqrt[x^7*(27*x + 4*K[4]^3)])^(1/3)*(6*2^(1/3)*Sqrt[3]*x^5*K[4] + 2*2^(1/3)*x*K[4]*Sqrt[x^7*(27*x + 4*K[4]^3)] + 3*(3*I + Sqrt[3])*x^4*(-27*x^4 - 2*x^3*K[4]^3 + 3*Sqrt[3]*Sqrt[x^7*(27*x + 4*K[4]^3)])^(1/3) + (1 + I*Sqrt[3])*Sqrt[x^7*(27*x + 4*K[4]^3)]*(-27*x^4 - 2*x^3*K[4]^3 + 3*Sqrt[3]*Sqrt[x^7*(27*x + 4*K[4]^3)])^(1/3)))/((I + Sqrt[3])*x^2*K[4]^3*Sqrt[x^7*(27*x + 4*K[4]^3)]) - Inactive[Integrate][(3*K[3]^8*((-108*I)*2^(1/3)*Sqrt[3]*K[3]^5*K[4] + (36*I)*2^(1/3)*K[3]*K[4]*Sqrt[K[3]^7*(27*K[3] + 4*K[4]^3)] + (2*I)*(3*I + Sqrt[3])*K[3]^3*K[4]^3*(-27*K[3]^4 - 2*K[3]^3*K[4]^3 + 3*Sqrt[3]*Sqrt[K[3]^7*(27*K[3] + 4*K[4]^3)])^(1/3) - 9*(-I + Sqrt[3])*Sqrt[K[3]^7*(27*K[3] + 4*K[4]^3)]*(-27*K[3]^4 - 2*K[3]^3*K[4]^3 + 3*Sqrt[3]*Sqrt[K[3]^7*(27*K[3] + 4*K[4]^3)])^(1/3) + K[3]^4*((-4*I)*2^(1/3)*Sqrt[3]*K[4]^4 + 27*(3 - I*Sqrt[3])*(-27*K[3]^4 - 2*K[3]^3*K[4]^3 + 3*Sqrt[3]*Sqrt[K[3]^7*(27*K[3] + 4*K[4]^3)])^(1/3))))/((I + Sqrt[3])*(K[3]^7*(27*K[3] + 4*K[4]^3))^(3/2)*(-54*K[3]^4 - 4*K[3]^3*K[4]^3 + 6*Sqrt[3]*Sqrt[K[3]^7*(27*K[3] + 4*K[4]^3)])^(2/3)), {K[3], 1, x}], {K[4], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(108*(-I + Sqrt[3])*K[5]^6*y[x]^2 + 16*(-I + Sqrt[3])*K[5]^5*y[x]^5 + (8*I)*2^(2/3)*K[5]^4*y[x]^4*(-27*K[5]^4 - 2*K[5]^3*y[x]^3 + 3*Sqrt[3]*Sqrt[K[5]^7*(27*K[5] + 4*y[x]^3)])^(1/3) + (6*I)*2^(2/3)*K[5]*y[x]*(9*K[5]^4 + Sqrt[3]*Sqrt[K[5]^7*(27*K[5] + 4*y[x]^3)])*(-27*K[5]^4 - 2*K[5]^3*y[x]^3 + 3*Sqrt[3]*Sqrt[K[5]^7*(27*K[5] + 4*y[x]^3)])^(1/3) + 4*2^(1/3)*(I + Sqrt[3])*K[5]^3*y[x]^3*(-27*K[5]^4 - 2*K[5]^3*y[x]^3 + 3*Sqrt[3]*Sqrt[K[5]^7*(27*K[5] + 4*y[x]^3)])^(2/3) + 3*2^(1/3)*(-27*K[5]^4 - 2*K[5]^3*y[x]^3 + 3*Sqrt[3]*Sqrt[K[5]^7*(27*K[5] + 4*y[x]^3)])^(2/3)*(9*(I + Sqrt[3])*K[5]^4 + (3 + I*Sqrt[3])*Sqrt[K[5]^7*(27*K[5] + 4*y[x]^3)]))/(4*(-I + Sqrt[3])*K[5]^6*y[x]^2*(27*K[5] + 4*y[x]^3)), {K[5], 1, x}] + Inactive[Integrate][(-3*(-54*x^4 - 4*x^3*K[6]^3 + 6*Sqrt[3]*Sqrt[x^7*(27*x + 4*K[6]^3)])^(1/3)*((6*I)*2^(1/3)*Sqrt[3]*x^5*K[6] + (2*I)*2^(1/3)*x*K[6]*Sqrt[x^7*(27*x + 4*K[6]^3)] + 3*(3 + I*Sqrt[3])*x^4*(-27*x^4 - 2*x^3*K[6]^3 + 3*Sqrt[3]*Sqrt[x^7*(27*x + 4*K[6]^3)])^(1/3) + (I + Sqrt[3])*Sqrt[x^7*(27*x + 4*K[6]^3)]*(-27*x^4 - 2*x^3*K[6]^3 + 3*Sqrt[3]*Sqrt[x^7*(27*x + 4*K[6]^3)])^(1/3)))/(4*(-I + Sqrt[3])*x^2*K[6]^3*Sqrt[x^7*(27*x + 4*K[6]^3)]) - Inactive[Integrate][(3*K[5]^8*((108*I)*2^(1/3)*Sqrt[3]*K[5]^5*K[6] - (36*I)*2^(1/3)*K[5]*K[6]*Sqrt[K[5]^7*(27*K[5] + 4*K[6]^3)] - (2*I)*(-3*I + Sqrt[3])*K[5]^3*K[6]^3*(-27*K[5]^4 - 2*K[5]^3*K[6]^3 + 3*Sqrt[3]*Sqrt[K[5]^7*(27*K[5] + 4*K[6]^3)])^(1/3) - 9*(I + Sqrt[3])*Sqrt[K[5]^7*(27*K[5] + 4*K[6]^3)]*(-27*K[5]^4 - 2*K[5]^3*K[6]^3 + 3*Sqrt[3]*Sqrt[K[5]^7*(27*K[5] + 4*K[6]^3)])^(1/3) + K[5]^4*((4*I)*2^(1/3)*Sqrt[3]*K[6]^4 + 27*(3 + I*Sqrt[3])*(-27*K[5]^4 - 2*K[5]^3*K[6]^3 + 3*Sqrt[3]*Sqrt[K[5]^7*(27*K[5] + 4*K[6]^3)])^(1/3))))/((-I + Sqrt[3])*(K[5]^7*(27*K[5] + 4*K[6]^3))^(3/2)*(-54*K[5]^4 - 4*K[5]^3*K[6]^3 + 6*Sqrt[3]*Sqrt[K[5]^7*(27*K[5] + 4*K[6]^3)])^(2/3)), {K[5], 1, x}], {K[6], 1, y[x]}], y[x]]}
Maple raw input
dsolve(x^2*diff(y(x),x)^3-2*x*y(x)*diff(y(x),x)^2+y(x)^2*diff(y(x),x)+1 = 0, y(x))
Maple raw output
[y(x) = 3/2*(-2*x)^(1/3), y(x) = -3/4*(-2*x)^(1/3)-3/4*I*3^(1/2)*(-2*x)^(1/3), y(x) = -3/4*(-2*x)^(1/3)+3/4*I*3^(1/2)*(-2*x)^(1/3), y(x) = x*_C1-1/(-_C1)^(1/2), y(x) = x*_C1+1/(-_C1)^(1/2)]