ODE
\[ y(x)^4 y'(x)^3-6 x y'(x)+2 y(x)=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries]]
Book solution method
No Missing Variables ODE, Solve for \(x\)
Mathematica ✓
cpu = 9.01906 (sec), leaf count = 2057
\[\left \{\text {Solve}\left [c_1=\int _1^x-\frac {\sqrt [3]{\sqrt {y(x)^{18}-8 K[1]^3 y(x)^{12}}-y(x)^9} \left (\sqrt [3]{\sqrt {y(x)^{12} \left (y(x)^6-8 K[1]^3\right )}-y(x)^9} y(x)^{10}-8 K[1]^3 \sqrt [3]{\sqrt {y(x)^{12} \left (y(x)^6-8 K[1]^3\right )}-y(x)^9} y(x)^4+\sqrt {y(x)^{12} \left (y(x)^6-8 K[1]^3\right )} \sqrt [3]{\sqrt {y(x)^{12} \left (y(x)^6-8 K[1]^3\right )}-y(x)^9} y(x)+4 K[1]^2 \sqrt {y(x)^{18}-8 K[1]^3 y(x)^{12}}\right )}{2 K[1]^2 y(x)^8 \left (8 K[1]^3-y(x)^6\right )}dK[1]+\int _1^{y(x)}\frac {-2 x K[2]^{14}+x \left (K[2]^6-8 x^3\right ) \int _1^x\frac {4 K[2]^{16} \left (K[2]^{14}+4 K[1]^3 K[2]^8-\sqrt {K[2]^{12} \left (K[2]^6-8 K[1]^3\right )} K[2]^5-K[1] \left (\sqrt {K[2]^{12} \left (K[2]^6-8 K[1]^3\right )}-3 K[2]^9\right ) \sqrt [3]{\sqrt {K[2]^{12} \left (K[2]^6-8 K[1]^3\right )}-K[2]^9}\right )}{\left (K[2]^{12} \left (K[2]^6-8 K[1]^3\right )\right )^{3/2} \left (\sqrt {K[2]^{12} \left (K[2]^6-8 K[1]^3\right )}-K[2]^9\right )^{2/3}}dK[1] K[2]^9+16 x^4 K[2]^8-8 x^3 \left (\sqrt {K[2]^{12} \left (K[2]^6-8 x^3\right )}-K[2]^9\right )^{2/3} K[2]^4+\left (\sqrt {K[2]^{12} \left (K[2]^6-8 x^3\right )}-K[2]^9\right )^{2/3} \left (K[2]^9+\sqrt {K[2]^{12} \left (K[2]^6-8 x^3\right )}\right ) K[2]+4 x^2 \sqrt {K[2]^{18}-8 x^3 K[2]^{12}} \sqrt [3]{\sqrt {K[2]^{12} \left (K[2]^6-8 x^3\right )}-K[2]^9}}{K[2]^9 \left (8 x^4-x K[2]^6\right )}dK[2],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {\left (1-i \sqrt {3}\right ) \left (\sqrt {y(x)^{12} \left (y(x)^6-8 K[3]^3\right )}-y(x)^9\right )^{2/3} y(x)^{10}+8 i \left (i+\sqrt {3}\right ) K[3]^3 \left (\sqrt {y(x)^{12} \left (y(x)^6-8 K[3]^3\right )}-y(x)^9\right )^{2/3} y(x)^4+\left (1-i \sqrt {3}\right ) \sqrt {y(x)^{12} \left (y(x)^6-8 K[3]^3\right )} \left (\sqrt {y(x)^{12} \left (y(x)^6-8 K[3]^3\right )}-y(x)^9\right )^{2/3} y(x)+4 \left (1+i \sqrt {3}\right ) K[3]^2 \sqrt {y(x)^{12} \left (y(x)^6-8 K[3]^3\right )} \sqrt [3]{\sqrt {y(x)^{12} \left (y(x)^6-8 K[3]^3\right )}-y(x)^9}}{4 K[3]^2 y(x)^8 \left (8 K[3]^3-y(x)^6\right )}dK[3]+\int _1^{y(x)}\frac {4 x K[4]^{14}+2 x \left (8 x^3-K[4]^6\right ) \int _1^x\frac {2 K[4]^{16} \left (-4 i \left (-i+\sqrt {3}\right ) K[3]^3 K[4]^8+\left (1+i \sqrt {3}\right ) \left (\sqrt {K[4]^{12} \left (K[4]^6-8 K[3]^3\right )}-K[4]^9\right ) K[4]^5+\left (1-i \sqrt {3}\right ) K[3] \left (\sqrt {K[4]^{12} \left (K[4]^6-8 K[3]^3\right )}-3 K[4]^9\right ) \sqrt [3]{\sqrt {K[4]^{12} \left (K[4]^6-8 K[3]^3\right )}-K[4]^9}\right )}{\left (K[4]^{12} \left (K[4]^6-8 K[3]^3\right )\right )^{3/2} \left (\sqrt {K[4]^{12} \left (K[4]^6-8 K[3]^3\right )}-K[4]^9\right )^{2/3}}dK[3] K[4]^9-32 x^4 K[4]^8+8 i \left (i+\sqrt {3}\right ) x^3 \left (\sqrt {K[4]^{12} \left (K[4]^6-8 x^3\right )}-K[4]^9\right )^{2/3} K[4]^4+\left (1-i \sqrt {3}\right ) \left (\sqrt {K[4]^{12} \left (K[4]^6-8 x^3\right )}-K[4]^9\right )^{2/3} \left (K[4]^9+\sqrt {K[4]^{12} \left (K[4]^6-8 x^3\right )}\right ) K[4]+4 \left (1+i \sqrt {3}\right ) x^2 \sqrt {K[4]^{12} \left (K[4]^6-8 x^3\right )} \sqrt [3]{\sqrt {K[4]^{12} \left (K[4]^6-8 x^3\right )}-K[4]^9}}{2 x K[4]^9 \left (K[4]^6-8 x^3\right )}dK[4],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {\left (1+i \sqrt {3}\right ) \left (\sqrt {y(x)^{12} \left (y(x)^6-8 K[5]^3\right )}-y(x)^9\right )^{2/3} y(x)^{10}-8 i \left (-i+\sqrt {3}\right ) K[5]^3 \left (\sqrt {y(x)^{12} \left (y(x)^6-8 K[5]^3\right )}-y(x)^9\right )^{2/3} y(x)^4+\left (1+i \sqrt {3}\right ) \sqrt {y(x)^{12} \left (y(x)^6-8 K[5]^3\right )} \left (\sqrt {y(x)^{12} \left (y(x)^6-8 K[5]^3\right )}-y(x)^9\right )^{2/3} y(x)+4 \left (1-i \sqrt {3}\right ) K[5]^2 \sqrt {y(x)^{12} \left (y(x)^6-8 K[5]^3\right )} \sqrt [3]{\sqrt {y(x)^{12} \left (y(x)^6-8 K[5]^3\right )}-y(x)^9}}{4 K[5]^2 y(x)^8 \left (8 K[5]^3-y(x)^6\right )}dK[5]+\int _1^{y(x)}\frac {4 x K[6]^{14}+2 x \left (8 x^3-K[6]^6\right ) \int _1^x\frac {2 K[6]^{16} \left (4 i \left (i+\sqrt {3}\right ) K[5]^3 K[6]^8+i \left (i+\sqrt {3}\right ) \left (K[6]^9-\sqrt {K[6]^{12} \left (K[6]^6-8 K[5]^3\right )}\right ) K[6]^5+\left (1+i \sqrt {3}\right ) K[5] \left (\sqrt {K[6]^{12} \left (K[6]^6-8 K[5]^3\right )}-3 K[6]^9\right ) \sqrt [3]{\sqrt {K[6]^{12} \left (K[6]^6-8 K[5]^3\right )}-K[6]^9}\right )}{\left (K[6]^{12} \left (K[6]^6-8 K[5]^3\right )\right )^{3/2} \left (\sqrt {K[6]^{12} \left (K[6]^6-8 K[5]^3\right )}-K[6]^9\right )^{2/3}}dK[5] K[6]^9-32 x^4 K[6]^8-8 i \left (-i+\sqrt {3}\right ) x^3 \left (\sqrt {K[6]^{12} \left (K[6]^6-8 x^3\right )}-K[6]^9\right )^{2/3} K[6]^4+\left (1+i \sqrt {3}\right ) \left (\sqrt {K[6]^{12} \left (K[6]^6-8 x^3\right )}-K[6]^9\right )^{2/3} \left (K[6]^9+\sqrt {K[6]^{12} \left (K[6]^6-8 x^3\right )}\right ) K[6]+4 \left (1-i \sqrt {3}\right ) x^2 \sqrt {K[6]^{12} \left (K[6]^6-8 x^3\right )} \sqrt [3]{\sqrt {K[6]^{12} \left (K[6]^6-8 x^3\right )}-K[6]^9}}{2 x K[6]^9 \left (K[6]^6-8 x^3\right )}dK[6],y(x)\right ]\right \}\]
Maple ✓
cpu = 0.976 (sec), leaf count = 179
\[\left [y \left (x \right ) = \sqrt {2}\, \sqrt {x}, y \left (x \right ) = -\sqrt {2}\, \sqrt {x}, y \left (x \right ) = \sqrt {-i \sqrt {3}\, x -x}, y \left (x \right ) = \sqrt {i \sqrt {3}\, x -x}, y \left (x \right ) = -\sqrt {-i \sqrt {3}\, x -x}, y \left (x \right ) = -\sqrt {i \sqrt {3}\, x -x}, y \left (x \right ) = \frac {\left (-4 \textit {\_C1}^{3}+24 x \textit {\_C1} \right )^{\frac {1}{3}}}{2}, y \left (x \right ) = -\frac {\left (-4 \textit {\_C1}^{3}+24 x \textit {\_C1} \right )^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, \left (-4 \textit {\_C1}^{3}+24 x \textit {\_C1} \right )^{\frac {1}{3}}}{4}, y \left (x \right ) = -\frac {\left (-4 \textit {\_C1}^{3}+24 x \textit {\_C1} \right )^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, \left (-4 \textit {\_C1}^{3}+24 x \textit {\_C1} \right )^{\frac {1}{3}}}{4}\right ]\] Mathematica raw input
DSolve[2*y[x] - 6*x*y'[x] + y[x]^4*y'[x]^3 == 0,y[x],x]
Mathematica raw output
{Solve[C[1] == Inactive[Integrate][-1/2*((-y[x]^9 + Sqrt[-8*K[1]^3*y[x]^12 + y[x]^18])^(1/3)*(4*K[1]^2*Sqrt[-8*K[1]^3*y[x]^12 + y[x]^18] - 8*K[1]^3*y[x]^4*(-y[x]^9 + Sqrt[y[x]^12*(-8*K[1]^3 + y[x]^6)])^(1/3) + y[x]^10*(-y[x]^9 + Sqrt[y[x]^12*(-8*K[1]^3 + y[x]^6)])^(1/3) + y[x]*Sqrt[y[x]^12*(-8*K[1]^3 + y[x]^6)]*(-y[x]^9 + Sqrt[y[x]^12*(-8*K[1]^3 + y[x]^6)])^(1/3)))/(K[1]^2*y[x]^8*(8*K[1]^3 - y[x]^6)), {K[1], 1, x}] + Inactive[Integrate][(16*x^4*K[2]^8 - 2*x*K[2]^14 + 4*x^2*Sqrt[-8*x^3*K[2]^12 + K[2]^18]*(-K[2]^9 + Sqrt[K[2]^12*(-8*x^3 + K[2]^6)])^(1/3) - 8*x^3*K[2]^4*(-K[2]^9 + Sqrt[K[2]^12*(-8*x^3 + K[2]^6)])^(2/3) + K[2]*(-K[2]^9 + Sqrt[K[2]^12*(-8*x^3 + K[2]^6)])^(2/3)*(K[2]^9 + Sqrt[K[2]^12*(-8*x^3 + K[2]^6)]) + x*K[2]^9*(-8*x^3 + K[2]^6)*Inactive[Integrate][(4*K[2]^16*(4*K[1]^3*K[2]^8 + K[2]^14 - K[2]^5*Sqrt[K[2]^12*(-8*K[1]^3 + K[2]^6)] - K[1]*(-3*K[2]^9 + Sqrt[K[2]^12*(-8*K[1]^3 + K[2]^6)])*(-K[2]^9 + Sqrt[K[2]^12*(-8*K[1]^3 + K[2]^6)])^(1/3)))/((K[2]^12*(-8*K[1]^3 + K[2]^6))^(3/2)*(-K[2]^9 + Sqrt[K[2]^12*(-8*K[1]^3 + K[2]^6)])^(2/3)), {K[1], 1, x}])/(K[2]^9*(8*x^4 - x*K[2]^6)), {K[2], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(4*(1 + I*Sqrt[3])*K[3]^2*Sqrt[y[x]^12*(-8*K[3]^3 + y[x]^6)]*(-y[x]^9 + Sqrt[y[x]^12*(-8*K[3]^3 + y[x]^6)])^(1/3) + (8*I)*(I + Sqrt[3])*K[3]^3*y[x]^4*(-y[x]^9 + Sqrt[y[x]^12*(-8*K[3]^3 + y[x]^6)])^(2/3) + (1 - I*Sqrt[3])*y[x]^10*(-y[x]^9 + Sqrt[y[x]^12*(-8*K[3]^3 + y[x]^6)])^(2/3) + (1 - I*Sqrt[3])*y[x]*Sqrt[y[x]^12*(-8*K[3]^3 + y[x]^6)]*(-y[x]^9 + Sqrt[y[x]^12*(-8*K[3]^3 + y[x]^6)])^(2/3))/(4*K[3]^2*y[x]^8*(8*K[3]^3 - y[x]^6)), {K[3], 1, x}] + Inactive[Integrate][(-32*x^4*K[4]^8 + 4*x*K[4]^14 + 4*(1 + I*Sqrt[3])*x^2*Sqrt[K[4]^12*(-8*x^3 + K[4]^6)]*(-K[4]^9 + Sqrt[K[4]^12*(-8*x^3 + K[4]^6)])^(1/3) + (8*I)*(I + Sqrt[3])*x^3*K[4]^4*(-K[4]^9 + Sqrt[K[4]^12*(-8*x^3 + K[4]^6)])^(2/3) + (1 - I*Sqrt[3])*K[4]*(-K[4]^9 + Sqrt[K[4]^12*(-8*x^3 + K[4]^6)])^(2/3)*(K[4]^9 + Sqrt[K[4]^12*(-8*x^3 + K[4]^6)]) + 2*x*K[4]^9*(8*x^3 - K[4]^6)*Inactive[Integrate][(2*K[4]^16*((-4*I)*(-I + Sqrt[3])*K[3]^3*K[4]^8 + (1 - I*Sqrt[3])*K[3]*(-3*K[4]^9 + Sqrt[K[4]^12*(-8*K[3]^3 + K[4]^6)])*(-K[4]^9 + Sqrt[K[4]^12*(-8*K[3]^3 + K[4]^6)])^(1/3) + (1 + I*Sqrt[3])*K[4]^5*(-K[4]^9 + Sqrt[K[4]^12*(-8*K[3]^3 + K[4]^6)])))/((K[4]^12*(-8*K[3]^3 + K[4]^6))^(3/2)*(-K[4]^9 + Sqrt[K[4]^12*(-8*K[3]^3 + K[4]^6)])^(2/3)), {K[3], 1, x}])/(2*x*K[4]^9*(-8*x^3 + K[4]^6)), {K[4], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(4*(1 - I*Sqrt[3])*K[5]^2*Sqrt[y[x]^12*(-8*K[5]^3 + y[x]^6)]*(-y[x]^9 + Sqrt[y[x]^12*(-8*K[5]^3 + y[x]^6)])^(1/3) - (8*I)*(-I + Sqrt[3])*K[5]^3*y[x]^4*(-y[x]^9 + Sqrt[y[x]^12*(-8*K[5]^3 + y[x]^6)])^(2/3) + (1 + I*Sqrt[3])*y[x]^10*(-y[x]^9 + Sqrt[y[x]^12*(-8*K[5]^3 + y[x]^6)])^(2/3) + (1 + I*Sqrt[3])*y[x]*Sqrt[y[x]^12*(-8*K[5]^3 + y[x]^6)]*(-y[x]^9 + Sqrt[y[x]^12*(-8*K[5]^3 + y[x]^6)])^(2/3))/(4*K[5]^2*y[x]^8*(8*K[5]^3 - y[x]^6)), {K[5], 1, x}] + Inactive[Integrate][(-32*x^4*K[6]^8 + 4*x*K[6]^14 + 4*(1 - I*Sqrt[3])*x^2*Sqrt[K[6]^12*(-8*x^3 + K[6]^6)]*(-K[6]^9 + Sqrt[K[6]^12*(-8*x^3 + K[6]^6)])^(1/3) - (8*I)*(-I + Sqrt[3])*x^3*K[6]^4*(-K[6]^9 + Sqrt[K[6]^12*(-8*x^3 + K[6]^6)])^(2/3) + (1 + I*Sqrt[3])*K[6]*(-K[6]^9 + Sqrt[K[6]^12*(-8*x^3 + K[6]^6)])^(2/3)*(K[6]^9 + Sqrt[K[6]^12*(-8*x^3 + K[6]^6)]) + 2*x*K[6]^9*(8*x^3 - K[6]^6)*Inactive[Integrate][(2*K[6]^16*((4*I)*(I + Sqrt[3])*K[5]^3*K[6]^8 + I*(I + Sqrt[3])*K[6]^5*(K[6]^9 - Sqrt[K[6]^12*(-8*K[5]^3 + K[6]^6)]) + (1 + I*Sqrt[3])*K[5]*(-3*K[6]^9 + Sqrt[K[6]^12*(-8*K[5]^3 + K[6]^6)])*(-K[6]^9 + Sqrt[K[6]^12*(-8*K[5]^3 + K[6]^6)])^(1/3)))/((K[6]^12*(-8*K[5]^3 + K[6]^6))^(3/2)*(-K[6]^9 + Sqrt[K[6]^12*(-8*K[5]^3 + K[6]^6)])^(2/3)), {K[5], 1, x}])/(2*x*K[6]^9*(-8*x^3 + K[6]^6)), {K[6], 1, y[x]}], y[x]]}
Maple raw input
dsolve(y(x)^4*diff(y(x),x)^3-6*x*diff(y(x),x)+2*y(x) = 0, y(x))
Maple raw output
[y(x) = 2^(1/2)*x^(1/2), y(x) = -2^(1/2)*x^(1/2), y(x) = (-I*x*3^(1/2)-x)^(1/2), y(x) = (I*x*3^(1/2)-x)^(1/2), y(x) = -(-I*x*3^(1/2)-x)^(1/2), y(x) = -(I*x*3^(1/2)-x)^(1/2), y(x) = 1/2*(-4*_C1^3+24*_C1*x)^(1/3), y(x) = -1/4*(-4*_C1^3+24*_C1*x)^(1/3)-1/4*I*3^(1/2)*(-4*_C1^3+24*_C1*x)^(1/3), y(x) = -1/4*(-4*_C1^3+24*_C1*x)^(1/3)+1/4*I*3^(1/2)*(-4*_C1^3+24*_C1*x)^(1/3)]