ODE
\[ x^4 \left (-y'(x)\right )+2 x^3 y(x)+3 y'(x)^3=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries]]
Book solution method
No Missing Variables ODE, Solve for \(y\)
Mathematica ✓
cpu = 10.0662 (sec), leaf count = 2375
\[\left \{\text {Solve}\left [c_1=\int _1^x\frac {\left (9 y(x) K[1]^3+\sqrt {-K[1]^6 \left (K[1]^6-81 y(x)^2\right )}\right )^{2/3} K[1]^9-\sqrt {-K[1]^6 \left (K[1]^6-81 y(x)^2\right )} \sqrt [3]{9 y(x) K[1]^3+\sqrt {-K[1]^6 \left (K[1]^6-81 y(x)^2\right )}} K[1]^5-81 y(x)^2 \left (9 y(x) K[1]^3+\sqrt {-K[1]^6 \left (K[1]^6-81 y(x)^2\right )}\right )^{2/3} K[1]^3+9 y(x) \sqrt {K[1]^6 \left (81 y(x)^2-K[1]^6\right )} \left (9 y(x) K[1]^3+\sqrt {K[1]^6 \left (81 y(x)^2-K[1]^6\right )}\right )^{2/3}}{K[1]^8 \left (K[1]^6-81 y(x)^2\right )}dK[1]+\int _1^{y(x)}\frac {x^{13}-\left (9 K[2] x^3+\sqrt {-x^6 \left (x^6-81 K[2]^2\right )}\right )^{2/3} x^9-81 K[2]^2 x^7-3 K[2] \left (x^6-81 K[2]^2\right ) \int _1^x-\frac {3 K[1]^9 \left (K[1]^9-27 K[2] \sqrt [3]{9 K[2] K[1]^3+\sqrt {-K[1]^6 \left (K[1]^6-81 K[2]^2\right )}} K[1]^4+162 K[2]^2 K[1]^3-\sqrt {-K[1]^6 \left (K[1]^6-81 K[2]^2\right )} \sqrt [3]{9 K[2] K[1]^3+\sqrt {-K[1]^6 \left (K[1]^6-81 K[2]^2\right )}} K[1]+18 K[2] \sqrt {-K[1]^6 \left (K[1]^6-81 K[2]^2\right )}\right )}{\left (-K[1]^6 \left (K[1]^6-81 K[2]^2\right )\right )^{3/2} \left (9 K[2] K[1]^3+\sqrt {-K[1]^6 \left (K[1]^6-81 K[2]^2\right )}\right )^{2/3}}dK[1] x^7+\sqrt {-x^6 \left (x^6-81 K[2]^2\right )} \sqrt [3]{9 K[2] x^3+\sqrt {-x^6 \left (x^6-81 K[2]^2\right )}} x^5+81 K[2]^2 \left (9 K[2] x^3+\sqrt {-x^6 \left (x^6-81 K[2]^2\right )}\right )^{2/3} x^3-9 K[2] \sqrt {x^6 \left (81 K[2]^2-x^6\right )} \left (9 K[2] x^3+\sqrt {x^6 \left (81 K[2]^2-x^6\right )}\right )^{2/3}}{3 x^7 K[2] \left (x^6-81 K[2]^2\right )}dK[2],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {\left (-1-i \sqrt {3}\right ) \left (9 y(x) K[3]^3+\sqrt {-K[3]^6 \left (K[3]^6-81 y(x)^2\right )}\right )^{2/3} K[3]^9+\left (1-i \sqrt {3}\right ) \sqrt {-K[3]^6 \left (K[3]^6-81 y(x)^2\right )} \sqrt [3]{9 y(x) K[3]^3+\sqrt {-K[3]^6 \left (K[3]^6-81 y(x)^2\right )}} K[3]^5+81 \left (1+i \sqrt {3}\right ) y(x)^2 \left (9 y(x) K[3]^3+\sqrt {-K[3]^6 \left (K[3]^6-81 y(x)^2\right )}\right )^{2/3} K[3]^3-9 i \left (-i+\sqrt {3}\right ) y(x) \sqrt {-K[3]^6 \left (K[3]^6-81 y(x)^2\right )} \left (9 y(x) K[3]^3+\sqrt {-K[3]^6 \left (K[3]^6-81 y(x)^2\right )}\right )^{2/3}}{2 K[3]^8 \left (K[3]^6-81 y(x)^2\right )}dK[3]+\int _1^{y(x)}\frac {2 x^{13}+\left (1+i \sqrt {3}\right ) \left (9 K[4] x^3+\sqrt {-x^6 \left (x^6-81 K[4]^2\right )}\right )^{2/3} x^9-162 K[4]^2 x^7-6 K[4] \left (x^6-81 K[4]^2\right ) \int _1^x-\frac {3 K[3]^9 \left (i \left (i+\sqrt {3}\right ) K[3]^9+27 \left (1+i \sqrt {3}\right ) K[4] \sqrt [3]{9 K[4] K[3]^3+\sqrt {-K[3]^6 \left (K[3]^6-81 K[4]^2\right )}} K[3]^4+162 i \left (i+\sqrt {3}\right ) K[4]^2 K[3]^3+\left (1+i \sqrt {3}\right ) \sqrt {-K[3]^6 \left (K[3]^6-81 K[4]^2\right )} \sqrt [3]{9 K[4] K[3]^3+\sqrt {-K[3]^6 \left (K[3]^6-81 K[4]^2\right )}} K[3]+18 i \left (i+\sqrt {3}\right ) K[4] \sqrt {-K[3]^6 \left (K[3]^6-81 K[4]^2\right )}\right )}{2 \left (-K[3]^6 \left (K[3]^6-81 K[4]^2\right )\right )^{3/2} \left (9 K[4] K[3]^3+\sqrt {-K[3]^6 \left (K[3]^6-81 K[4]^2\right )}\right )^{2/3}}dK[3] x^7+i \left (i+\sqrt {3}\right ) \sqrt {-x^6 \left (x^6-81 K[4]^2\right )} \sqrt [3]{9 K[4] x^3+\sqrt {-x^6 \left (x^6-81 K[4]^2\right )}} x^5-81 i \left (-i+\sqrt {3}\right ) K[4]^2 \left (9 K[4] x^3+\sqrt {-x^6 \left (x^6-81 K[4]^2\right )}\right )^{2/3} x^3+9 \left (1+i \sqrt {3}\right ) K[4] \sqrt {-x^6 \left (x^6-81 K[4]^2\right )} \left (9 K[4] x^3+\sqrt {-x^6 \left (x^6-81 K[4]^2\right )}\right )^{2/3}}{6 x^7 K[4] \left (x^6-81 K[4]^2\right )}dK[4],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {i \left (i+\sqrt {3}\right ) \left (9 y(x) K[5]^3+\sqrt {-K[5]^6 \left (K[5]^6-81 y(x)^2\right )}\right )^{2/3} K[5]^9+\left (1+i \sqrt {3}\right ) \sqrt {-K[5]^6 \left (K[5]^6-81 y(x)^2\right )} \sqrt [3]{9 y(x) K[5]^3+\sqrt {-K[5]^6 \left (K[5]^6-81 y(x)^2\right )}} K[5]^5+81 \left (1-i \sqrt {3}\right ) y(x)^2 \left (9 y(x) K[5]^3+\sqrt {-K[5]^6 \left (K[5]^6-81 y(x)^2\right )}\right )^{2/3} K[5]^3+9 i \left (i+\sqrt {3}\right ) y(x) \sqrt {-K[5]^6 \left (K[5]^6-81 y(x)^2\right )} \left (9 y(x) K[5]^3+\sqrt {-K[5]^6 \left (K[5]^6-81 y(x)^2\right )}\right )^{2/3}}{2 K[5]^8 \left (K[5]^6-81 y(x)^2\right )}dK[5]+\int _1^{y(x)}-\frac {-2 x^{13}+i \left (i+\sqrt {3}\right ) \left (9 K[6] x^3+\sqrt {-x^6 \left (x^6-81 K[6]^2\right )}\right )^{2/3} x^9+162 K[6]^2 x^7+6 K[6] \left (x^6-81 K[6]^2\right ) \int _1^x-\frac {3 K[5]^9 \left (\left (-1-i \sqrt {3}\right ) K[5]^9+27 \left (1-i \sqrt {3}\right ) K[6] \sqrt [3]{9 K[6] K[5]^3+\sqrt {-K[5]^6 \left (K[5]^6-81 K[6]^2\right )}} K[5]^4-162 i \left (-i+\sqrt {3}\right ) K[6]^2 K[5]^3+\left (1-i \sqrt {3}\right ) \sqrt {-K[5]^6 \left (K[5]^6-81 K[6]^2\right )} \sqrt [3]{9 K[6] K[5]^3+\sqrt {-K[5]^6 \left (K[5]^6-81 K[6]^2\right )}} K[5]-18 i \left (-i+\sqrt {3}\right ) K[6] \sqrt {-K[5]^6 \left (K[5]^6-81 K[6]^2\right )}\right )}{2 \left (-K[5]^6 \left (K[5]^6-81 K[6]^2\right )\right )^{3/2} \left (9 K[6] K[5]^3+\sqrt {-K[5]^6 \left (K[5]^6-81 K[6]^2\right )}\right )^{2/3}}dK[5] x^7+\left (1+i \sqrt {3}\right ) \sqrt {-x^6 \left (x^6-81 K[6]^2\right )} \sqrt [3]{9 K[6] x^3+\sqrt {-x^6 \left (x^6-81 K[6]^2\right )}} x^5+81 \left (1-i \sqrt {3}\right ) K[6]^2 \left (9 K[6] x^3+\sqrt {-x^6 \left (x^6-81 K[6]^2\right )}\right )^{2/3} x^3+9 i \left (i+\sqrt {3}\right ) K[6] \sqrt {-x^6 \left (x^6-81 K[6]^2\right )} \left (9 K[6] x^3+\sqrt {-x^6 \left (x^6-81 K[6]^2\right )}\right )^{2/3}}{6 x^7 K[6] \left (x^6-81 K[6]^2\right )}dK[6],y(x)\right ]\right \}\]
Maple ✓
cpu = 1.118 (sec), leaf count = 34
\[\left [y \left (x \right ) = -\frac {x^{3}}{9}, y \left (x \right ) = \frac {x^{3}}{9}, y \left (x \right ) = \frac {x^{2}}{2 \textit {\_C1}}-\frac {3}{2 \textit {\_C1}^{3}}\right ]\] Mathematica raw input
DSolve[2*x^3*y[x] - x^4*y'[x] + 3*y'[x]^3 == 0,y[x],x]
Mathematica raw output
{Solve[C[1] == Inactive[Integrate][(-(K[1]^5*Sqrt[-(K[1]^6*(K[1]^6 - 81*y[x]^2))]*(9*K[1]^3*y[x] + Sqrt[-(K[1]^6*(K[1]^6 - 81*y[x]^2))])^(1/3)) + K[1]^9*(9*K[1]^3*y[x] + Sqrt[-(K[1]^6*(K[1]^6 - 81*y[x]^2))])^(2/3) - 81*K[1]^3*y[x]^2*(9*K[1]^3*y[x] + Sqrt[-(K[1]^6*(K[1]^6 - 81*y[x]^2))])^(2/3) + 9*y[x]*Sqrt[K[1]^6*(-K[1]^6 + 81*y[x]^2)]*(9*K[1]^3*y[x] + Sqrt[K[1]^6*(-K[1]^6 + 81*y[x]^2)])^(2/3))/(K[1]^8*(K[1]^6 - 81*y[x]^2)), {K[1], 1, x}] + Inactive[Integrate][(x^13 - 81*x^7*K[2]^2 + x^5*Sqrt[-(x^6*(x^6 - 81*K[2]^2))]*(9*x^3*K[2] + Sqrt[-(x^6*(x^6 - 81*K[2]^2))])^(1/3) - x^9*(9*x^3*K[2] + Sqrt[-(x^6*(x^6 - 81*K[2]^2))])^(2/3) + 81*x^3*K[2]^2*(9*x^3*K[2] + Sqrt[-(x^6*(x^6 - 81*K[2]^2))])^(2/3) - 9*K[2]*Sqrt[x^6*(-x^6 + 81*K[2]^2)]*(9*x^3*K[2] + Sqrt[x^6*(-x^6 + 81*K[2]^2)])^(2/3) - 3*x^7*K[2]*(x^6 - 81*K[2]^2)*Inactive[Integrate][(-3*K[1]^9*(K[1]^9 + 162*K[1]^3*K[2]^2 + 18*K[2]*Sqrt[-(K[1]^6*(K[1]^6 - 81*K[2]^2))] - 27*K[1]^4*K[2]*(9*K[1]^3*K[2] + Sqrt[-(K[1]^6*(K[1]^6 - 81*K[2]^2))])^(1/3) - K[1]*Sqrt[-(K[1]^6*(K[1]^6 - 81*K[2]^2))]*(9*K[1]^3*K[2] + Sqrt[-(K[1]^6*(K[1]^6 - 81*K[2]^2))])^(1/3)))/((-(K[1]^6*(K[1]^6 - 81*K[2]^2)))^(3/2)*(9*K[1]^3*K[2] + Sqrt[-(K[1]^6*(K[1]^6 - 81*K[2]^2))])^(2/3)), {K[1], 1, x}])/(3*x^7*K[2]*(x^6 - 81*K[2]^2)), {K[2], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][((1 - I*Sqrt[3])*K[3]^5*Sqrt[-(K[3]^6*(K[3]^6 - 81*y[x]^2))]*(9*K[3]^3*y[x] + Sqrt[-(K[3]^6*(K[3]^6 - 81*y[x]^2))])^(1/3) + (-1 - I*Sqrt[3])*K[3]^9*(9*K[3]^3*y[x] + Sqrt[-(K[3]^6*(K[3]^6 - 81*y[x]^2))])^(2/3) + 81*(1 + I*Sqrt[3])*K[3]^3*y[x]^2*(9*K[3]^3*y[x] + Sqrt[-(K[3]^6*(K[3]^6 - 81*y[x]^2))])^(2/3) - (9*I)*(-I + Sqrt[3])*y[x]*Sqrt[-(K[3]^6*(K[3]^6 - 81*y[x]^2))]*(9*K[3]^3*y[x] + Sqrt[-(K[3]^6*(K[3]^6 - 81*y[x]^2))])^(2/3))/(2*K[3]^8*(K[3]^6 - 81*y[x]^2)), {K[3], 1, x}] + Inactive[Integrate][(2*x^13 - 162*x^7*K[4]^2 + I*(I + Sqrt[3])*x^5*Sqrt[-(x^6*(x^6 - 81*K[4]^2))]*(9*x^3*K[4] + Sqrt[-(x^6*(x^6 - 81*K[4]^2))])^(1/3) + (1 + I*Sqrt[3])*x^9*(9*x^3*K[4] + Sqrt[-(x^6*(x^6 - 81*K[4]^2))])^(2/3) - (81*I)*(-I + Sqrt[3])*x^3*K[4]^2*(9*x^3*K[4] + Sqrt[-(x^6*(x^6 - 81*K[4]^2))])^(2/3) + 9*(1 + I*Sqrt[3])*K[4]*Sqrt[-(x^6*(x^6 - 81*K[4]^2))]*(9*x^3*K[4] + Sqrt[-(x^6*(x^6 - 81*K[4]^2))])^(2/3) - 6*x^7*K[4]*(x^6 - 81*K[4]^2)*Inactive[Integrate][(-3*K[3]^9*(I*(I + Sqrt[3])*K[3]^9 + (162*I)*(I + Sqrt[3])*K[3]^3*K[4]^2 + (18*I)*(I + Sqrt[3])*K[4]*Sqrt[-(K[3]^6*(K[3]^6 - 81*K[4]^2))] + 27*(1 + I*Sqrt[3])*K[3]^4*K[4]*(9*K[3]^3*K[4] + Sqrt[-(K[3]^6*(K[3]^6 - 81*K[4]^2))])^(1/3) + (1 + I*Sqrt[3])*K[3]*Sqrt[-(K[3]^6*(K[3]^6 - 81*K[4]^2))]*(9*K[3]^3*K[4] + Sqrt[-(K[3]^6*(K[3]^6 - 81*K[4]^2))])^(1/3)))/(2*(-(K[3]^6*(K[3]^6 - 81*K[4]^2)))^(3/2)*(9*K[3]^3*K[4] + Sqrt[-(K[3]^6*(K[3]^6 - 81*K[4]^2))])^(2/3)), {K[3], 1, x}])/(6*x^7*K[4]*(x^6 - 81*K[4]^2)), {K[4], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][((1 + I*Sqrt[3])*K[5]^5*Sqrt[-(K[5]^6*(K[5]^6 - 81*y[x]^2))]*(9*K[5]^3*y[x] + Sqrt[-(K[5]^6*(K[5]^6 - 81*y[x]^2))])^(1/3) + I*(I + Sqrt[3])*K[5]^9*(9*K[5]^3*y[x] + Sqrt[-(K[5]^6*(K[5]^6 - 81*y[x]^2))])^(2/3) + 81*(1 - I*Sqrt[3])*K[5]^3*y[x]^2*(9*K[5]^3*y[x] + Sqrt[-(K[5]^6*(K[5]^6 - 81*y[x]^2))])^(2/3) + (9*I)*(I + Sqrt[3])*y[x]*Sqrt[-(K[5]^6*(K[5]^6 - 81*y[x]^2))]*(9*K[5]^3*y[x] + Sqrt[-(K[5]^6*(K[5]^6 - 81*y[x]^2))])^(2/3))/(2*K[5]^8*(K[5]^6 - 81*y[x]^2)), {K[5], 1, x}] + Inactive[Integrate][-1/6*(-2*x^13 + 162*x^7*K[6]^2 + (1 + I*Sqrt[3])*x^5*Sqrt[-(x^6*(x^6 - 81*K[6]^2))]*(9*x^3*K[6] + Sqrt[-(x^6*(x^6 - 81*K[6]^2))])^(1/3) + I*(I + Sqrt[3])*x^9*(9*x^3*K[6] + Sqrt[-(x^6*(x^6 - 81*K[6]^2))])^(2/3) + 81*(1 - I*Sqrt[3])*x^3*K[6]^2*(9*x^3*K[6] + Sqrt[-(x^6*(x^6 - 81*K[6]^2))])^(2/3) + (9*I)*(I + Sqrt[3])*K[6]*Sqrt[-(x^6*(x^6 - 81*K[6]^2))]*(9*x^3*K[6] + Sqrt[-(x^6*(x^6 - 81*K[6]^2))])^(2/3) + 6*x^7*K[6]*(x^6 - 81*K[6]^2)*Inactive[Integrate][(-3*K[5]^9*((-1 - I*Sqrt[3])*K[5]^9 - (162*I)*(-I + Sqrt[3])*K[5]^3*K[6]^2 - (18*I)*(-I + Sqrt[3])*K[6]*Sqrt[-(K[5]^6*(K[5]^6 - 81*K[6]^2))] + 27*(1 - I*Sqrt[3])*K[5]^4*K[6]*(9*K[5]^3*K[6] + Sqrt[-(K[5]^6*(K[5]^6 - 81*K[6]^2))])^(1/3) + (1 - I*Sqrt[3])*K[5]*Sqrt[-(K[5]^6*(K[5]^6 - 81*K[6]^2))]*(9*K[5]^3*K[6] + Sqrt[-(K[5]^6*(K[5]^6 - 81*K[6]^2))])^(1/3)))/(2*(-(K[5]^6*(K[5]^6 - 81*K[6]^2)))^(3/2)*(9*K[5]^3*K[6] + Sqrt[-(K[5]^6*(K[5]^6 - 81*K[6]^2))])^(2/3)), {K[5], 1, x}])/(x^7*K[6]*(x^6 - 81*K[6]^2)), {K[6], 1, y[x]}], y[x]]}
Maple raw input
dsolve(3*diff(y(x),x)^3-x^4*diff(y(x),x)+2*x^3*y(x) = 0, y(x))
Maple raw output
[y(x) = -1/9*x^3, y(x) = 1/9*x^3, y(x) = 1/2/_C1*x^2-3/2/_C1^3]