ODE
\[ 2 x y'(x)^3-3 y(x) y'(x)^2-x=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries], _dAlembert]
Book solution method
Clairaut’s equation and related types, d’Alembert’s equation (also call Lagrange’s)
Mathematica ✓
cpu = 9.23303 (sec), leaf count = 2211
\[\left \{\text {Solve}\left [c_1=\int _1^x\frac {y(x)^5+\sqrt [3]{y(x)^3+2 \left (K[1]^3+\sqrt {K[1]^3 \left (K[1]^3+y(x)^3\right )}\right )} y(x)^4+\left (y(x)^3+2 \left (K[1]^3+\sqrt {K[1]^3 \left (K[1]^3+y(x)^3\right )}\right )\right )^{2/3} y(x)^3+K[1]^3 y(x)^2+\left (K[1]^3-\sqrt {K[1]^3 \left (K[1]^3+y(x)^3\right )}\right ) \sqrt [3]{y(x)^3+2 \left (K[1]^3+\sqrt {K[1]^3 \left (K[1]^3+y(x)^3\right )}\right )} y(x)+\left (K[1]^3-\sqrt {K[1]^3 \left (K[1]^3+y(x)^3\right )}\right ) \left (y(x)^3+2 \left (K[1]^3+\sqrt {K[1]^3 \left (K[1]^3+y(x)^3\right )}\right )\right )^{2/3}}{K[1] y(x)^2 \left (K[1]^3+y(x)^3\right )}dK[1]+\int _1^{y(x)}\left (\frac {\left (-x^3-K[2]^3+\sqrt {x^3 \left (x^3+K[2]^3\right )}\right ) \sqrt [3]{2 x^3+K[2]^3+2 \sqrt {x^3 \left (x^3+K[2]^3\right )}} \left (K[2]+\sqrt [3]{2 x^3+K[2]^3+2 \sqrt {x^3 \left (x^3+K[2]^3\right )}}\right )}{K[2]^3 \left (x^3+K[2]^3\right )}-\int _1^x\frac {K[1]^5 \left (K[2]^4-\sqrt [3]{2 K[1]^3+K[2]^3+2 \sqrt {K[1]^3 \left (K[1]^3+K[2]^3\right )}} K[2]^3+4 \sqrt {K[1]^3 \left (K[1]^3+K[2]^3\right )} K[2]+2 K[1]^3 \left (2 K[2]+\sqrt [3]{2 K[1]^3+K[2]^3+2 \sqrt {K[1]^3 \left (K[1]^3+K[2]^3\right )}}\right )+2 \sqrt {K[1]^3 \left (K[1]^3+K[2]^3\right )} \sqrt [3]{2 K[1]^3+K[2]^3+2 \sqrt {K[1]^3 \left (K[1]^3+K[2]^3\right )}}\right )}{2 \left (K[1]^3 \left (K[1]^3+K[2]^3\right )\right )^{3/2} \left (2 K[1]^3+K[2]^3+2 \sqrt {K[1]^3 \left (K[1]^3+K[2]^3\right )}\right )^{2/3}}dK[1]\right )dK[2],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {\left (-i+\sqrt {3}\right ) y(x)^5-\left (i+\sqrt {3}\right ) \sqrt [3]{y(x)^3+2 \left (K[3]^3+\sqrt {K[3]^3 \left (K[3]^3+y(x)^3\right )}\right )} y(x)^4+2 i \left (y(x)^3+2 \left (K[3]^3+\sqrt {K[3]^3 \left (K[3]^3+y(x)^3\right )}\right )\right )^{2/3} y(x)^3+\left (-i+\sqrt {3}\right ) K[3]^3 y(x)^2+\left (i+\sqrt {3}\right ) \left (\sqrt {K[3]^3 \left (K[3]^3+y(x)^3\right )}-K[3]^3\right ) \sqrt [3]{y(x)^3+2 \left (K[3]^3+\sqrt {K[3]^3 \left (K[3]^3+y(x)^3\right )}\right )} y(x)-2 i \left (\sqrt {K[3]^3 \left (K[3]^3+y(x)^3\right )}-K[3]^3\right ) \left (y(x)^3+2 \left (K[3]^3+\sqrt {K[3]^3 \left (K[3]^3+y(x)^3\right )}\right )\right )^{2/3}}{\left (-i+\sqrt {3}\right ) K[3] y(x)^2 (K[3]+y(x)) \left (K[3]^2-y(x) K[3]+y(x)^2\right )}dK[3]+\int _1^{y(x)}-\frac {\left (-i+\sqrt {3}\right ) \left (x^3+K[4]^3\right ) \int _1^x-\frac {K[3]^2 \left (\left (i+\sqrt {3}\right ) K[4]^4+2 i \sqrt [3]{2 K[3]^3+K[4]^3+2 \sqrt {K[3]^3 \left (K[3]^3+K[4]^3\right )}} K[4]^3+4 \left (i+\sqrt {3}\right ) \sqrt {K[3]^3 \left (K[3]^3+K[4]^3\right )} K[4]+4 K[3]^3 \left (\left (i+\sqrt {3}\right ) K[4]-i \sqrt [3]{2 K[3]^3+K[4]^3+2 \sqrt {K[3]^3 \left (K[3]^3+K[4]^3\right )}}\right )-4 i \sqrt {K[3]^3 \left (K[3]^3+K[4]^3\right )} \sqrt [3]{2 K[3]^3+K[4]^3+2 \sqrt {K[3]^3 \left (K[3]^3+K[4]^3\right )}}\right )}{2 \left (-i+\sqrt {3}\right ) (K[3]+K[4]) \left (K[3]^2-K[4] K[3]+K[4]^2\right ) \sqrt {K[3]^3 \left (K[3]^3+K[4]^3\right )} \left (2 K[3]^3+K[4]^3+2 \sqrt {K[3]^3 \left (K[3]^3+K[4]^3\right )}\right )^{2/3}}dK[3] K[4]^3+\left (-x^3-K[4]^3+\sqrt {x^3 \left (x^3+K[4]^3\right )}\right ) \sqrt [3]{2 x^3+K[4]^3+2 \sqrt {x^3 \left (x^3+K[4]^3\right )}} \left (\left (i+\sqrt {3}\right ) K[4]-2 i \sqrt [3]{2 x^3+K[4]^3+2 \sqrt {x^3 \left (x^3+K[4]^3\right )}}\right )}{\left (-i+\sqrt {3}\right ) K[4]^3 (x+K[4]) \left (x^2-K[4] x+K[4]^2\right )}dK[4],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {\left (i+\sqrt {3}\right ) y(x)^5-\left (-i+\sqrt {3}\right ) \sqrt [3]{y(x)^3+2 \left (K[5]^3+\sqrt {K[5]^3 \left (K[5]^3+y(x)^3\right )}\right )} y(x)^4-2 i \left (y(x)^3+2 \left (K[5]^3+\sqrt {K[5]^3 \left (K[5]^3+y(x)^3\right )}\right )\right )^{2/3} y(x)^3+\left (i+\sqrt {3}\right ) K[5]^3 y(x)^2+\left (-i+\sqrt {3}\right ) \left (\sqrt {K[5]^3 \left (K[5]^3+y(x)^3\right )}-K[5]^3\right ) \sqrt [3]{y(x)^3+2 \left (K[5]^3+\sqrt {K[5]^3 \left (K[5]^3+y(x)^3\right )}\right )} y(x)+2 i \left (\sqrt {K[5]^3 \left (K[5]^3+y(x)^3\right )}-K[5]^3\right ) \left (y(x)^3+2 \left (K[5]^3+\sqrt {K[5]^3 \left (K[5]^3+y(x)^3\right )}\right )\right )^{2/3}}{\left (i+\sqrt {3}\right ) K[5] y(x)^2 (K[5]+y(x)) \left (K[5]^2-y(x) K[5]+y(x)^2\right )}dK[5]+\int _1^{y(x)}-\frac {\left (i+\sqrt {3}\right ) \left (x^3+K[6]^3\right ) \int _1^x-\frac {K[5]^2 \left (\left (-i+\sqrt {3}\right ) K[6]^4-2 i \sqrt [3]{2 K[5]^3+K[6]^3+2 \sqrt {K[5]^3 \left (K[5]^3+K[6]^3\right )}} K[6]^3+4 \left (-i+\sqrt {3}\right ) \sqrt {K[5]^3 \left (K[5]^3+K[6]^3\right )} K[6]+4 K[5]^3 \left (\left (-i+\sqrt {3}\right ) K[6]+i \sqrt [3]{2 K[5]^3+K[6]^3+2 \sqrt {K[5]^3 \left (K[5]^3+K[6]^3\right )}}\right )+4 i \sqrt {K[5]^3 \left (K[5]^3+K[6]^3\right )} \sqrt [3]{2 K[5]^3+K[6]^3+2 \sqrt {K[5]^3 \left (K[5]^3+K[6]^3\right )}}\right )}{2 \left (i+\sqrt {3}\right ) (K[5]+K[6]) \left (K[5]^2-K[6] K[5]+K[6]^2\right ) \sqrt {K[5]^3 \left (K[5]^3+K[6]^3\right )} \left (2 K[5]^3+K[6]^3+2 \sqrt {K[5]^3 \left (K[5]^3+K[6]^3\right )}\right )^{2/3}}dK[5] K[6]^3+\left (-x^3-K[6]^3+\sqrt {x^3 \left (x^3+K[6]^3\right )}\right ) \sqrt [3]{2 x^3+K[6]^3+2 \sqrt {x^3 \left (x^3+K[6]^3\right )}} \left (\left (-i+\sqrt {3}\right ) K[6]+2 i \sqrt [3]{2 x^3+K[6]^3+2 \sqrt {x^3 \left (x^3+K[6]^3\right )}}\right )}{\left (i+\sqrt {3}\right ) K[6]^3 (x+K[6]) \left (x^2-K[6] x+K[6]^2\right )}dK[6],y(x)\right ]\right \}\]
Maple ✓
cpu = 0.079 (sec), leaf count = 69
\[\left [y \left (x \right ) = -x, y \left (x \right ) = \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x, y \left (x \right ) = \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x, y \left (x \right ) = -\frac {\left (-\frac {2 \left (x \textit {\_C1} \right )^{\frac {3}{2}}}{\textit {\_C1}^{3}}+1\right ) \textit {\_C1}}{3}, y \left (x \right ) = -\frac {\left (\frac {2 \left (x \textit {\_C1} \right )^{\frac {3}{2}}}{\textit {\_C1}^{3}}+1\right ) \textit {\_C1}}{3}\right ]\] Mathematica raw input
DSolve[-x - 3*y[x]*y'[x]^2 + 2*x*y'[x]^3 == 0,y[x],x]
Mathematica raw output
{Solve[C[1] == Inactive[Integrate][(K[1]^3*y[x]^2 + y[x]^5 + y[x]^4*(y[x]^3 + 2*(K[1]^3 + Sqrt[K[1]^3*(K[1]^3 + y[x]^3)]))^(1/3) + y[x]*(K[1]^3 - Sqrt[K[1]^3*(K[1]^3 + y[x]^3)])*(y[x]^3 + 2*(K[1]^3 + Sqrt[K[1]^3*(K[1]^3 + y[x]^3)]))^(1/3) + y[x]^3*(y[x]^3 + 2*(K[1]^3 + Sqrt[K[1]^3*(K[1]^3 + y[x]^3)]))^(2/3) + (K[1]^3 - Sqrt[K[1]^3*(K[1]^3 + y[x]^3)])*(y[x]^3 + 2*(K[1]^3 + Sqrt[K[1]^3*(K[1]^3 + y[x]^3)]))^(2/3))/(K[1]*y[x]^2*(K[1]^3 + y[x]^3)), {K[1], 1, x}] + Inactive[Integrate][((-x^3 - K[2]^3 + Sqrt[x^3*(x^3 + K[2]^3)])*(2*x^3 + K[2]^3 + 2*Sqrt[x^3*(x^3 + K[2]^3)])^(1/3)*(K[2] + (2*x^3 + K[2]^3 + 2*Sqrt[x^3*(x^3 + K[2]^3)])^(1/3)))/(K[2]^3*(x^3 + K[2]^3)) - Inactive[Integrate][(K[1]^5*(K[2]^4 + 4*K[2]*Sqrt[K[1]^3*(K[1]^3 + K[2]^3)] - K[2]^3*(2*K[1]^3 + K[2]^3 + 2*Sqrt[K[1]^3*(K[1]^3 + K[2]^3)])^(1/3) + 2*Sqrt[K[1]^3*(K[1]^3 + K[2]^3)]*(2*K[1]^3 + K[2]^3 + 2*Sqrt[K[1]^3*(K[1]^3 + K[2]^3)])^(1/3) + 2*K[1]^3*(2*K[2] + (2*K[1]^3 + K[2]^3 + 2*Sqrt[K[1]^3*(K[1]^3 + K[2]^3)])^(1/3))))/(2*(K[1]^3*(K[1]^3 + K[2]^3))^(3/2)*(2*K[1]^3 + K[2]^3 + 2*Sqrt[K[1]^3*(K[1]^3 + K[2]^3)])^(2/3)), {K[1], 1, x}], {K[2], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][((-I + Sqrt[3])*K[3]^3*y[x]^2 + (-I + Sqrt[3])*y[x]^5 - (I + Sqrt[3])*y[x]^4*(y[x]^3 + 2*(K[3]^3 + Sqrt[K[3]^3*(K[3]^3 + y[x]^3)]))^(1/3) + (I + Sqrt[3])*y[x]*(-K[3]^3 + Sqrt[K[3]^3*(K[3]^3 + y[x]^3)])*(y[x]^3 + 2*(K[3]^3 + Sqrt[K[3]^3*(K[3]^3 + y[x]^3)]))^(1/3) + (2*I)*y[x]^3*(y[x]^3 + 2*(K[3]^3 + Sqrt[K[3]^3*(K[3]^3 + y[x]^3)]))^(2/3) - (2*I)*(-K[3]^3 + Sqrt[K[3]^3*(K[3]^3 + y[x]^3)])*(y[x]^3 + 2*(K[3]^3 + Sqrt[K[3]^3*(K[3]^3 + y[x]^3)]))^(2/3))/((-I + Sqrt[3])*K[3]*y[x]^2*(K[3] + y[x])*(K[3]^2 - K[3]*y[x] + y[x]^2)), {K[3], 1, x}] + Inactive[Integrate][-(((-x^3 - K[4]^3 + Sqrt[x^3*(x^3 + K[4]^3)])*(2*x^3 + K[4]^3 + 2*Sqrt[x^3*(x^3 + K[4]^3)])^(1/3)*((I + Sqrt[3])*K[4] - (2*I)*(2*x^3 + K[4]^3 + 2*Sqrt[x^3*(x^3 + K[4]^3)])^(1/3)) + (-I + Sqrt[3])*K[4]^3*(x^3 + K[4]^3)*Inactive[Integrate][-1/2*(K[3]^2*((I + Sqrt[3])*K[4]^4 + 4*(I + Sqrt[3])*K[4]*Sqrt[K[3]^3*(K[3]^3 + K[4]^3)] + (2*I)*K[4]^3*(2*K[3]^3 + K[4]^3 + 2*Sqrt[K[3]^3*(K[3]^3 + K[4]^3)])^(1/3) - (4*I)*Sqrt[K[3]^3*(K[3]^3 + K[4]^3)]*(2*K[3]^3 + K[4]^3 + 2*Sqrt[K[3]^3*(K[3]^3 + K[4]^3)])^(1/3) + 4*K[3]^3*((I + Sqrt[3])*K[4] - I*(2*K[3]^3 + K[4]^3 + 2*Sqrt[K[3]^3*(K[3]^3 + K[4]^3)])^(1/3))))/((-I + Sqrt[3])*(K[3] + K[4])*(K[3]^2 - K[3]*K[4] + K[4]^2)*Sqrt[K[3]^3*(K[3]^3 + K[4]^3)]*(2*K[3]^3 + K[4]^3 + 2*Sqrt[K[3]^3*(K[3]^3 + K[4]^3)])^(2/3)), {K[3], 1, x}])/((-I + Sqrt[3])*K[4]^3*(x + K[4])*(x^2 - x*K[4] + K[4]^2))), {K[4], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][((I + Sqrt[3])*K[5]^3*y[x]^2 + (I + Sqrt[3])*y[x]^5 - (-I + Sqrt[3])*y[x]^4*(y[x]^3 + 2*(K[5]^3 + Sqrt[K[5]^3*(K[5]^3 + y[x]^3)]))^(1/3) + (-I + Sqrt[3])*y[x]*(-K[5]^3 + Sqrt[K[5]^3*(K[5]^3 + y[x]^3)])*(y[x]^3 + 2*(K[5]^3 + Sqrt[K[5]^3*(K[5]^3 + y[x]^3)]))^(1/3) - (2*I)*y[x]^3*(y[x]^3 + 2*(K[5]^3 + Sqrt[K[5]^3*(K[5]^3 + y[x]^3)]))^(2/3) + (2*I)*(-K[5]^3 + Sqrt[K[5]^3*(K[5]^3 + y[x]^3)])*(y[x]^3 + 2*(K[5]^3 + Sqrt[K[5]^3*(K[5]^3 + y[x]^3)]))^(2/3))/((I + Sqrt[3])*K[5]*y[x]^2*(K[5] + y[x])*(K[5]^2 - K[5]*y[x] + y[x]^2)), {K[5], 1, x}] + Inactive[Integrate][-(((-x^3 - K[6]^3 + Sqrt[x^3*(x^3 + K[6]^3)])*(2*x^3 + K[6]^3 + 2*Sqrt[x^3*(x^3 + K[6]^3)])^(1/3)*((-I + Sqrt[3])*K[6] + (2*I)*(2*x^3 + K[6]^3 + 2*Sqrt[x^3*(x^3 + K[6]^3)])^(1/3)) + (I + Sqrt[3])*K[6]^3*(x^3 + K[6]^3)*Inactive[Integrate][-1/2*(K[5]^2*((-I + Sqrt[3])*K[6]^4 + 4*(-I + Sqrt[3])*K[6]*Sqrt[K[5]^3*(K[5]^3 + K[6]^3)] - (2*I)*K[6]^3*(2*K[5]^3 + K[6]^3 + 2*Sqrt[K[5]^3*(K[5]^3 + K[6]^3)])^(1/3) + (4*I)*Sqrt[K[5]^3*(K[5]^3 + K[6]^3)]*(2*K[5]^3 + K[6]^3 + 2*Sqrt[K[5]^3*(K[5]^3 + K[6]^3)])^(1/3) + 4*K[5]^3*((-I + Sqrt[3])*K[6] + I*(2*K[5]^3 + K[6]^3 + 2*Sqrt[K[5]^3*(K[5]^3 + K[6]^3)])^(1/3))))/((I + Sqrt[3])*(K[5] + K[6])*(K[5]^2 - K[5]*K[6] + K[6]^2)*Sqrt[K[5]^3*(K[5]^3 + K[6]^3)]*(2*K[5]^3 + K[6]^3 + 2*Sqrt[K[5]^3*(K[5]^3 + K[6]^3)])^(2/3)), {K[5], 1, x}])/((I + Sqrt[3])*K[6]^3*(x + K[6])*(x^2 - x*K[6] + K[6]^2))), {K[6], 1, y[x]}], y[x]]}
Maple raw input
dsolve(2*x*diff(y(x),x)^3-3*y(x)*diff(y(x),x)^2-x = 0, y(x))
Maple raw output
[y(x) = -x, y(x) = (1/2-1/2*I*3^(1/2))*x, y(x) = (1/2+1/2*I*3^(1/2))*x, y(x) = -1/3*(-2/_C1^3*(x*_C1)^(3/2)+1)*_C1, y(x) = -1/3*(2/_C1^3*(x*_C1)^(3/2)+1)*_C1]