ODE
\[ 4 x^2+x y'(x)^3-2 y(x) y'(x)^2=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries]]
Book solution method
No Missing Variables ODE, Solve for \(y\)
Mathematica ✓
cpu = 72.7563 (sec), leaf count = 2824
\[\left \{\text {Solve}\left [c_1=\int _1^x\frac {-32 y(x)^5-16 \sqrt [3]{-54 K[1]^4+8 y(x)^3+6 \sqrt {81 K[1]^8-24 K[1]^4 y(x)^3}} y(x)^4-8 \left (-54 K[1]^4+8 y(x)^3+6 \sqrt {81 K[1]^8-24 K[1]^4 y(x)^3}\right )^{2/3} y(x)^3+108 K[1]^4 y(x)^2+6 \left (9 K[1]^4+\sqrt {81 K[1]^8-24 K[1]^4 y(x)^3}\right ) \sqrt [3]{-54 K[1]^4+8 y(x)^3+6 \sqrt {81 K[1]^8-24 K[1]^4 y(x)^3}} y(x)+3 \left (9 K[1]^4+\sqrt {81 K[1]^8-24 K[1]^4 y(x)^3}\right ) \left (-54 K[1]^4+8 y(x)^3+6 \sqrt {81 K[1]^8-24 K[1]^4 y(x)^3}\right )^{2/3}}{4 y(x)^2 \left (27 K[1]^5-8 K[1] y(x)^3\right )}dK[1]+\int _1^{y(x)}\frac {3 \left (27 x^4-8 K[2]^3+3 \sqrt {81 x^8-24 x^4 K[2]^3}\right ) \sqrt [3]{-54 x^4+8 K[2]^3+6 \sqrt {81 x^8-24 x^4 K[2]^3}} \left (2 K[2]+\sqrt [3]{-54 x^4+8 K[2]^3+6 \sqrt {81 x^8-24 x^4 K[2]^3}}\right )+16 \left (27 x^4 K[2]^3-8 K[2]^6\right ) \int _1^x\frac {3 \sqrt [3]{2} K[1]^3 \left (-27 \sqrt {3} \left (4 K[2]+\sqrt [3]{-54 K[1]^4+8 K[2]^3+6 \sqrt {81 K[1]^8-24 K[1]^4 K[2]^3}}\right ) K[1]^4+8 \sqrt {3} K[2]^4+36 K[2] \sqrt {27 K[1]^8-8 K[1]^4 K[2]^3}-4 \sqrt {3} K[2]^3 \sqrt [3]{-54 K[1]^4+8 K[2]^3+6 \sqrt {81 K[1]^8-24 K[1]^4 K[2]^3}}+9 \sqrt {27 K[1]^8-8 K[1]^4 K[2]^3} \sqrt [3]{-54 K[1]^4+8 K[2]^3+6 \sqrt {81 K[1]^8-24 K[1]^4 K[2]^3}}\right )}{\left (27 K[1]^4-8 K[2]^3\right ) \sqrt {27 K[1]^8-8 K[1]^4 K[2]^3} \left (-27 K[1]^4+4 K[2]^3+3 \sqrt {81 K[1]^8-24 K[1]^4 K[2]^3}\right )^{2/3}}dK[1]}{16 K[2]^3 \left (8 K[2]^3-27 x^4\right )}dK[2],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {-32 \left (i+\sqrt {3}\right ) y(x)^5+32 i \sqrt [3]{-54 K[3]^4+8 y(x)^3+6 \sqrt {81 K[3]^8-24 K[3]^4 y(x)^3}} y(x)^4+8 \left (-i+\sqrt {3}\right ) \left (-54 K[3]^4+8 y(x)^3+6 \sqrt {81 K[3]^8-24 K[3]^4 y(x)^3}\right )^{2/3} y(x)^3+108 \left (i+\sqrt {3}\right ) K[3]^4 y(x)^2-12 i \left (9 K[3]^4+\sqrt {81 K[3]^8-24 K[3]^4 y(x)^3}\right ) \sqrt [3]{-54 K[3]^4+8 y(x)^3+6 \sqrt {81 K[3]^8-24 K[3]^4 y(x)^3}} y(x)+3 i \left (-54 K[3]^4+8 y(x)^3+6 \sqrt {81 K[3]^8-24 K[3]^4 y(x)^3}\right )^{2/3} \left (\left (9+9 i \sqrt {3}\right ) K[3]^4+\left (3 i+\sqrt {3}\right ) \sqrt {27 K[3]^8-8 K[3]^4 y(x)^3}\right )}{4 \left (i+\sqrt {3}\right ) K[3] y(x)^2 \left (27 K[3]^4-8 y(x)^3\right )}dK[3]+\int _1^{y(x)}\frac {16 \left (i+\sqrt {3}\right ) \left (27 x^4-8 K[4]^3\right ) \int _1^x\frac {3 K[3]^3 \left (27 \left (8 i \sqrt [3]{2} \sqrt {3} K[4]+2^{2/3} \left (3-i \sqrt {3}\right ) \sqrt [3]{-27 K[3]^4+4 K[4]^3+3 \sqrt {81 K[3]^8-24 K[3]^4 K[4]^3}}\right ) K[3]^4-16 i \sqrt [3]{2} \sqrt {3} K[4]^4-72 i \sqrt [3]{2} K[4] \sqrt {27 K[3]^8-8 K[3]^4 K[4]^3}+4\ 2^{2/3} \left (3-i \sqrt {3}\right ) K[4]^3 \sqrt [3]{-27 K[3]^4+4 K[4]^3+3 \sqrt {81 K[3]^8-24 K[3]^4 K[4]^3}}-9\ 2^{2/3} \left (-i+\sqrt {3}\right ) \sqrt {27 K[3]^8-8 K[3]^4 K[4]^3} \sqrt [3]{-27 K[3]^4+4 K[4]^3+3 \sqrt {81 K[3]^8-24 K[3]^4 K[4]^3}}\right )}{\left (i+\sqrt {3}\right ) \left (27 K[3]^4-8 K[4]^3\right ) \sqrt {27 K[3]^8-8 K[3]^4 K[4]^3} \left (-27 K[3]^4+4 K[4]^3+3 \sqrt {81 K[3]^8-24 K[3]^4 K[4]^3}\right )^{2/3}}dK[3] K[4]^3+3 \sqrt [3]{-54 x^4+8 K[4]^3+6 \sqrt {81 x^8-24 x^4 K[4]^3}} \left (-27 \left (4 i K[4]+\left (-i+\sqrt {3}\right ) \sqrt [3]{-54 x^4+8 K[4]^3+6 \sqrt {81 x^8-24 x^4 K[4]^3}}\right ) x^4+32 i K[4]^4-12 i K[4] \sqrt {81 x^8-24 x^4 K[4]^3}+8 \left (-i+\sqrt {3}\right ) K[4]^3 \sqrt [3]{-54 x^4+8 K[4]^3+6 \sqrt {81 x^8-24 x^4 K[4]^3}}+3 i \left (3 i+\sqrt {3}\right ) \sqrt {27 x^8-8 x^4 K[4]^3} \sqrt [3]{-54 x^4+8 K[4]^3+6 \sqrt {81 x^8-24 x^4 K[4]^3}}\right )}{16 \left (i+\sqrt {3}\right ) K[4]^3 \left (8 K[4]^3-27 x^4\right )}dK[4],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {-32 \left (-i+\sqrt {3}\right ) y(x)^5-32 i \sqrt [3]{-54 K[5]^4+8 y(x)^3+6 \sqrt {81 K[5]^8-24 K[5]^4 y(x)^3}} y(x)^4+8 \left (i+\sqrt {3}\right ) \left (-54 K[5]^4+8 y(x)^3+6 \sqrt {81 K[5]^8-24 K[5]^4 y(x)^3}\right )^{2/3} y(x)^3+108 \left (-i+\sqrt {3}\right ) K[5]^4 y(x)^2+12 i \left (9 K[5]^4+\sqrt {81 K[5]^8-24 K[5]^4 y(x)^3}\right ) \sqrt [3]{-54 K[5]^4+8 y(x)^3+6 \sqrt {81 K[5]^8-24 K[5]^4 y(x)^3}} y(x)-3 i \left (-54 K[5]^4+8 y(x)^3+6 \sqrt {81 K[5]^8-24 K[5]^4 y(x)^3}\right )^{2/3} \left (\left (9-9 i \sqrt {3}\right ) K[5]^4+\left (-3 i+\sqrt {3}\right ) \sqrt {27 K[5]^8-8 K[5]^4 y(x)^3}\right )}{4 \left (-i+\sqrt {3}\right ) K[5] y(x)^2 \left (27 K[5]^4-8 y(x)^3\right )}dK[5]+\int _1^{y(x)}\left (\frac {3 \sqrt [3]{-54 x^4+8 K[6]^3+6 \sqrt {81 x^8-24 x^4 K[6]^3}} \left (-27 \left (\left (i+\sqrt {3}\right ) \sqrt [3]{-54 x^4+8 K[6]^3+6 \sqrt {81 x^8-24 x^4 K[6]^3}}-4 i K[6]\right ) x^4-32 i K[6]^4+12 i K[6] \sqrt {81 x^8-24 x^4 K[6]^3}+8 \left (i+\sqrt {3}\right ) K[6]^3 \sqrt [3]{-54 x^4+8 K[6]^3+6 \sqrt {81 x^8-24 x^4 K[6]^3}}-3 i \left (-3 i+\sqrt {3}\right ) \sqrt {27 x^8-8 x^4 K[6]^3} \sqrt [3]{-54 x^4+8 K[6]^3+6 \sqrt {81 x^8-24 x^4 K[6]^3}}\right )}{16 \left (-i+\sqrt {3}\right ) K[6]^3 \left (8 K[6]^3-27 x^4\right )}-\int _1^x\frac {3 K[5]^3 \left (27 \left (2^{2/3} \left (3+i \sqrt {3}\right ) \sqrt [3]{-27 K[5]^4+4 K[6]^3+3 \sqrt {81 K[5]^8-24 K[5]^4 K[6]^3}}-8 i \sqrt [3]{2} \sqrt {3} K[6]\right ) K[5]^4+16 i \sqrt [3]{2} \sqrt {3} K[6]^4+72 i \sqrt [3]{2} K[6] \sqrt {27 K[5]^8-8 K[5]^4 K[6]^3}+4\ 2^{2/3} \left (3+i \sqrt {3}\right ) K[6]^3 \sqrt [3]{-27 K[5]^4+4 K[6]^3+3 \sqrt {81 K[5]^8-24 K[5]^4 K[6]^3}}-9\ 2^{2/3} \left (i+\sqrt {3}\right ) \sqrt {27 K[5]^8-8 K[5]^4 K[6]^3} \sqrt [3]{-27 K[5]^4+4 K[6]^3+3 \sqrt {81 K[5]^8-24 K[5]^4 K[6]^3}}\right )}{\left (-i+\sqrt {3}\right ) \left (27 K[5]^4-8 K[6]^3\right ) \sqrt {27 K[5]^8-8 K[5]^4 K[6]^3} \left (-27 K[5]^4+4 K[6]^3+3 \sqrt {81 K[5]^8-24 K[5]^4 K[6]^3}\right )^{2/3}}dK[5]\right )dK[6],y(x)\right ]\right \}\]
Maple ✓
cpu = 2.704 (sec), leaf count = 422
\[\left [y \left (x \right ) = \frac {3 x^{\frac {4}{3}}}{2}, y \left (x \right ) = \frac {3 \left (-\frac {x^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, x^{\frac {1}{3}}}{2}\right ) x}{2}, y \left (x \right ) = \frac {3 \left (-\frac {x^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, x^{\frac {1}{3}}}{2}\right ) x}{2}, y \left (x \right )-\RootOf \left (-4 \,\mathrm {csgn}\left (\frac {\textit {\_C1}^{2}+32 \textit {\_Z}}{\textit {\_C1}}\right ) \sqrt {-2 \textit {\_C1}^{3}+64 \textit {\_C1} \textit {\_Z}}\, \textit {\_C1}^{3}-\mathrm {csgn}\left (\frac {\textit {\_C1}^{2}+32 \textit {\_Z}}{\textit {\_C1}}\right ) \left (-2 \textit {\_C1}^{3}+64 \textit {\_C1} \textit {\_Z} \right )^{\frac {3}{2}}+64 \textit {\_C1}^{3} x +\left (-2 \textit {\_C1}^{3}+64 \textit {\_C1} \textit {\_Z} \right )^{\frac {3}{2}}\right ) = 0, y \left (x \right )-\RootOf \left (4 \,\mathrm {csgn}\left (\frac {\textit {\_C1}^{2}+32 \textit {\_Z}}{\textit {\_C1}}\right ) \sqrt {-2 \textit {\_C1}^{3}+64 \textit {\_C1} \textit {\_Z}}\, \textit {\_C1}^{3}+\mathrm {csgn}\left (\frac {\textit {\_C1}^{2}+32 \textit {\_Z}}{\textit {\_C1}}\right ) \left (-2 \textit {\_C1}^{3}+64 \textit {\_C1} \textit {\_Z} \right )^{\frac {3}{2}}+64 \textit {\_C1}^{3} x -\left (-2 \textit {\_C1}^{3}+64 \textit {\_C1} \textit {\_Z} \right )^{\frac {3}{2}}\right ) = 0, y \left (x \right )-\RootOf \left (\textit {\_C1} \left (\sqrt {2 \textit {\_C1}^{3}-64 \textit {\_C1} \textit {\_Z}}\, \mathrm {csgn}\left (\frac {\textit {\_C1}^{2}+32 \textit {\_Z}}{\textit {\_C1}}\right ) \textit {\_C1}^{2}+32 \sqrt {2 \textit {\_C1}^{3}-64 \textit {\_C1} \textit {\_Z}}\, \mathrm {csgn}\left (\frac {\textit {\_C1}^{2}+32 \textit {\_Z}}{\textit {\_C1}}\right ) \textit {\_Z} +\sqrt {2 \textit {\_C1}^{3}-64 \textit {\_C1} \textit {\_Z}}\, \textit {\_C1}^{2}-32 \textit {\_C1}^{2} x -32 \sqrt {2 \textit {\_C1}^{3}-64 \textit {\_C1} \textit {\_Z}}\, \textit {\_Z} \right )\right ) = 0, y \left (x \right )-\RootOf \left (\textit {\_C1} \left (\sqrt {2 \textit {\_C1}^{3}-64 \textit {\_C1} \textit {\_Z}}\, \mathrm {csgn}\left (\frac {\textit {\_C1}^{2}+32 \textit {\_Z}}{\textit {\_C1}}\right ) \textit {\_C1}^{2}+32 \sqrt {2 \textit {\_C1}^{3}-64 \textit {\_C1} \textit {\_Z}}\, \mathrm {csgn}\left (\frac {\textit {\_C1}^{2}+32 \textit {\_Z}}{\textit {\_C1}}\right ) \textit {\_Z} +\sqrt {2 \textit {\_C1}^{3}-64 \textit {\_C1} \textit {\_Z}}\, \textit {\_C1}^{2}+32 \textit {\_C1}^{2} x -32 \sqrt {2 \textit {\_C1}^{3}-64 \textit {\_C1} \textit {\_Z}}\, \textit {\_Z} \right )\right ) = 0\right ]\] Mathematica raw input
DSolve[4*x^2 - 2*y[x]*y'[x]^2 + x*y'[x]^3 == 0,y[x],x]
Mathematica raw output
{Solve[C[1] == Inactive[Integrate][(108*K[1]^4*y[x]^2 - 32*y[x]^5 - 16*y[x]^4*(-54*K[1]^4 + 8*y[x]^3 + 6*Sqrt[81*K[1]^8 - 24*K[1]^4*y[x]^3])^(1/3) + 6*y[x]*(9*K[1]^4 + Sqrt[81*K[1]^8 - 24*K[1]^4*y[x]^3])*(-54*K[1]^4 + 8*y[x]^3 + 6*Sqrt[81*K[1]^8 - 24*K[1]^4*y[x]^3])^(1/3) - 8*y[x]^3*(-54*K[1]^4 + 8*y[x]^3 + 6*Sqrt[81*K[1]^8 - 24*K[1]^4*y[x]^3])^(2/3) + 3*(9*K[1]^4 + Sqrt[81*K[1]^8 - 24*K[1]^4*y[x]^3])*(-54*K[1]^4 + 8*y[x]^3 + 6*Sqrt[81*K[1]^8 - 24*K[1]^4*y[x]^3])^(2/3))/(4*y[x]^2*(27*K[1]^5 - 8*K[1]*y[x]^3)), {K[1], 1, x}] + Inactive[Integrate][(3*(27*x^4 - 8*K[2]^3 + 3*Sqrt[81*x^8 - 24*x^4*K[2]^3])*(-54*x^4 + 8*K[2]^3 + 6*Sqrt[81*x^8 - 24*x^4*K[2]^3])^(1/3)*(2*K[2] + (-54*x^4 + 8*K[2]^3 + 6*Sqrt[81*x^8 - 24*x^4*K[2]^3])^(1/3)) + 16*(27*x^4*K[2]^3 - 8*K[2]^6)*Inactive[Integrate][(3*2^(1/3)*K[1]^3*(8*Sqrt[3]*K[2]^4 + 36*K[2]*Sqrt[27*K[1]^8 - 8*K[1]^4*K[2]^3] - 4*Sqrt[3]*K[2]^3*(-54*K[1]^4 + 8*K[2]^3 + 6*Sqrt[81*K[1]^8 - 24*K[1]^4*K[2]^3])^(1/3) + 9*Sqrt[27*K[1]^8 - 8*K[1]^4*K[2]^3]*(-54*K[1]^4 + 8*K[2]^3 + 6*Sqrt[81*K[1]^8 - 24*K[1]^4*K[2]^3])^(1/3) - 27*Sqrt[3]*K[1]^4*(4*K[2] + (-54*K[1]^4 + 8*K[2]^3 + 6*Sqrt[81*K[1]^8 - 24*K[1]^4*K[2]^3])^(1/3))))/((27*K[1]^4 - 8*K[2]^3)*Sqrt[27*K[1]^8 - 8*K[1]^4*K[2]^3]*(-27*K[1]^4 + 4*K[2]^3 + 3*Sqrt[81*K[1]^8 - 24*K[1]^4*K[2]^3])^(2/3)), {K[1], 1, x}])/(16*K[2]^3*(-27*x^4 + 8*K[2]^3)), {K[2], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(108*(I + Sqrt[3])*K[3]^4*y[x]^2 - 32*(I + Sqrt[3])*y[x]^5 + (32*I)*y[x]^4*(-54*K[3]^4 + 8*y[x]^3 + 6*Sqrt[81*K[3]^8 - 24*K[3]^4*y[x]^3])^(1/3) - (12*I)*y[x]*(9*K[3]^4 + Sqrt[81*K[3]^8 - 24*K[3]^4*y[x]^3])*(-54*K[3]^4 + 8*y[x]^3 + 6*Sqrt[81*K[3]^8 - 24*K[3]^4*y[x]^3])^(1/3) + 8*(-I + Sqrt[3])*y[x]^3*(-54*K[3]^4 + 8*y[x]^3 + 6*Sqrt[81*K[3]^8 - 24*K[3]^4*y[x]^3])^(2/3) + (3*I)*(-54*K[3]^4 + 8*y[x]^3 + 6*Sqrt[81*K[3]^8 - 24*K[3]^4*y[x]^3])^(2/3)*((9 + (9*I)*Sqrt[3])*K[3]^4 + (3*I + Sqrt[3])*Sqrt[27*K[3]^8 - 8*K[3]^4*y[x]^3]))/(4*(I + Sqrt[3])*K[3]*y[x]^2*(27*K[3]^4 - 8*y[x]^3)), {K[3], 1, x}] + Inactive[Integrate][(3*(-54*x^4 + 8*K[4]^3 + 6*Sqrt[81*x^8 - 24*x^4*K[4]^3])^(1/3)*((32*I)*K[4]^4 - (12*I)*K[4]*Sqrt[81*x^8 - 24*x^4*K[4]^3] + 8*(-I + Sqrt[3])*K[4]^3*(-54*x^4 + 8*K[4]^3 + 6*Sqrt[81*x^8 - 24*x^4*K[4]^3])^(1/3) + (3*I)*(3*I + Sqrt[3])*Sqrt[27*x^8 - 8*x^4*K[4]^3]*(-54*x^4 + 8*K[4]^3 + 6*Sqrt[81*x^8 - 24*x^4*K[4]^3])^(1/3) - 27*x^4*((4*I)*K[4] + (-I + Sqrt[3])*(-54*x^4 + 8*K[4]^3 + 6*Sqrt[81*x^8 - 24*x^4*K[4]^3])^(1/3))) + 16*(I + Sqrt[3])*K[4]^3*(27*x^4 - 8*K[4]^3)*Inactive[Integrate][(3*K[3]^3*((-16*I)*2^(1/3)*Sqrt[3]*K[4]^4 - (72*I)*2^(1/3)*K[4]*Sqrt[27*K[3]^8 - 8*K[3]^4*K[4]^3] + 4*2^(2/3)*(3 - I*Sqrt[3])*K[4]^3*(-27*K[3]^4 + 4*K[4]^3 + 3*Sqrt[81*K[3]^8 - 24*K[3]^4*K[4]^3])^(1/3) - 9*2^(2/3)*(-I + Sqrt[3])*Sqrt[27*K[3]^8 - 8*K[3]^4*K[4]^3]*(-27*K[3]^4 + 4*K[4]^3 + 3*Sqrt[81*K[3]^8 - 24*K[3]^4*K[4]^3])^(1/3) + 27*K[3]^4*((8*I)*2^(1/3)*Sqrt[3]*K[4] + 2^(2/3)*(3 - I*Sqrt[3])*(-27*K[3]^4 + 4*K[4]^3 + 3*Sqrt[81*K[3]^8 - 24*K[3]^4*K[4]^3])^(1/3))))/((I + Sqrt[3])*(27*K[3]^4 - 8*K[4]^3)*Sqrt[27*K[3]^8 - 8*K[3]^4*K[4]^3]*(-27*K[3]^4 + 4*K[4]^3 + 3*Sqrt[81*K[3]^8 - 24*K[3]^4*K[4]^3])^(2/3)), {K[3], 1, x}])/(16*(I + Sqrt[3])*K[4]^3*(-27*x^4 + 8*K[4]^3)), {K[4], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(108*(-I + Sqrt[3])*K[5]^4*y[x]^2 - 32*(-I + Sqrt[3])*y[x]^5 - (32*I)*y[x]^4*(-54*K[5]^4 + 8*y[x]^3 + 6*Sqrt[81*K[5]^8 - 24*K[5]^4*y[x]^3])^(1/3) + (12*I)*y[x]*(9*K[5]^4 + Sqrt[81*K[5]^8 - 24*K[5]^4*y[x]^3])*(-54*K[5]^4 + 8*y[x]^3 + 6*Sqrt[81*K[5]^8 - 24*K[5]^4*y[x]^3])^(1/3) + 8*(I + Sqrt[3])*y[x]^3*(-54*K[5]^4 + 8*y[x]^3 + 6*Sqrt[81*K[5]^8 - 24*K[5]^4*y[x]^3])^(2/3) - (3*I)*(-54*K[5]^4 + 8*y[x]^3 + 6*Sqrt[81*K[5]^8 - 24*K[5]^4*y[x]^3])^(2/3)*((9 - (9*I)*Sqrt[3])*K[5]^4 + (-3*I + Sqrt[3])*Sqrt[27*K[5]^8 - 8*K[5]^4*y[x]^3]))/(4*(-I + Sqrt[3])*K[5]*y[x]^2*(27*K[5]^4 - 8*y[x]^3)), {K[5], 1, x}] + Inactive[Integrate][(3*(-54*x^4 + 8*K[6]^3 + 6*Sqrt[81*x^8 - 24*x^4*K[6]^3])^(1/3)*((-32*I)*K[6]^4 + (12*I)*K[6]*Sqrt[81*x^8 - 24*x^4*K[6]^3] + 8*(I + Sqrt[3])*K[6]^3*(-54*x^4 + 8*K[6]^3 + 6*Sqrt[81*x^8 - 24*x^4*K[6]^3])^(1/3) - (3*I)*(-3*I + Sqrt[3])*Sqrt[27*x^8 - 8*x^4*K[6]^3]*(-54*x^4 + 8*K[6]^3 + 6*Sqrt[81*x^8 - 24*x^4*K[6]^3])^(1/3) - 27*x^4*((-4*I)*K[6] + (I + Sqrt[3])*(-54*x^4 + 8*K[6]^3 + 6*Sqrt[81*x^8 - 24*x^4*K[6]^3])^(1/3))))/(16*(-I + Sqrt[3])*K[6]^3*(-27*x^4 + 8*K[6]^3)) - Inactive[Integrate][(3*K[5]^3*((16*I)*2^(1/3)*Sqrt[3]*K[6]^4 + (72*I)*2^(1/3)*K[6]*Sqrt[27*K[5]^8 - 8*K[5]^4*K[6]^3] + 4*2^(2/3)*(3 + I*Sqrt[3])*K[6]^3*(-27*K[5]^4 + 4*K[6]^3 + 3*Sqrt[81*K[5]^8 - 24*K[5]^4*K[6]^3])^(1/3) - 9*2^(2/3)*(I + Sqrt[3])*Sqrt[27*K[5]^8 - 8*K[5]^4*K[6]^3]*(-27*K[5]^4 + 4*K[6]^3 + 3*Sqrt[81*K[5]^8 - 24*K[5]^4*K[6]^3])^(1/3) + 27*K[5]^4*((-8*I)*2^(1/3)*Sqrt[3]*K[6] + 2^(2/3)*(3 + I*Sqrt[3])*(-27*K[5]^4 + 4*K[6]^3 + 3*Sqrt[81*K[5]^8 - 24*K[5]^4*K[6]^3])^(1/3))))/((-I + Sqrt[3])*(27*K[5]^4 - 8*K[6]^3)*Sqrt[27*K[5]^8 - 8*K[5]^4*K[6]^3]*(-27*K[5]^4 + 4*K[6]^3 + 3*Sqrt[81*K[5]^8 - 24*K[5]^4*K[6]^3])^(2/3)), {K[5], 1, x}], {K[6], 1, y[x]}], y[x]]}
Maple raw input
dsolve(x*diff(y(x),x)^3-2*y(x)*diff(y(x),x)^2+4*x^2 = 0, y(x))
Maple raw output
[y(x) = 3/2*x^(4/3), y(x) = 3/2*(-1/2*x^(1/3)-1/2*I*3^(1/2)*x^(1/3))*x, y(x) = 3/2*(-1/2*x^(1/3)+1/2*I*3^(1/2)*x^(1/3))*x, y(x)-RootOf(-4*csgn((_C1^2+32*_Z)/_C1)*(-2*_C1^3+64*_C1*_Z)^(1/2)*_C1^3-csgn((_C1^2+32*_Z)/_C1)*(-2*_C1^3+64*_C1*_Z)^(3/2)+64*_C1^3*x+(-2*_C1^3+64*_C1*_Z)^(3/2)) = 0, y(x)-RootOf(4*csgn((_C1^2+32*_Z)/_C1)*(-2*_C1^3+64*_C1*_Z)^(1/2)*_C1^3+csgn((_C1^2+32*_Z)/_C1)*(-2*_C1^3+64*_C1*_Z)^(3/2)+64*_C1^3*x-(-2*_C1^3+64*_C1*_Z)^(3/2)) = 0, y(x)-RootOf(_C1*((2*_C1^3-64*_C1*_Z)^(1/2)*csgn((_C1^2+32*_Z)/_C1)*_C1^2+32*(2*_C1^3-64*_C1*_Z)^(1/2)*csgn((_C1^2+32*_Z)/_C1)*_Z+(2*_C1^3-64*_C1*_Z)^(1/2)*_C1^2-32*_C1^2*x-32*(2*_C1^3-64*_C1*_Z)^(1/2)*_Z)) = 0, y(x)-RootOf(_C1*((2*_C1^3-64*_C1*_Z)^(1/2)*csgn((_C1^2+32*_Z)/_C1)*_C1^2+32*(2*_C1^3-64*_C1*_Z)^(1/2)*csgn((_C1^2+32*_Z)/_C1)*_Z+(2*_C1^3-64*_C1*_Z)^(1/2)*_C1^2+32*_C1^2*x-32*(2*_C1^3-64*_C1*_Z)^(1/2)*_Z)) = 0]