ODE
\[ -a x y(x) y'(x)+2 a y(x)^2+y'(x)^3=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries]]
Book solution method
No Missing Variables ODE, Solve for \(x\)
Mathematica ✓
cpu = 20.1898 (sec), leaf count = 2804
\[\left \{\text {Solve}\left [c_1=\int _1^x\frac {a^2 y(x) \left (3 \sqrt {3} \sqrt {-a^2 \left (a K[1]^3-27 y(x)\right ) y(x)^3}-27 a y(x)^2\right )^{2/3} K[1]^3-\sqrt [6]{3} \sqrt {-a^2 \left (a K[1]^3-27 y(x)\right ) y(x)^3} \sqrt [3]{\sqrt {3} \sqrt {-a^2 \left (a K[1]^3-27 y(x)\right ) y(x)^3}-9 a y(x)^2} \left (3^{2/3} a K[1]^2+9 \sqrt [3]{\sqrt {3} \sqrt {-a^2 \left (a K[1]^3-27 y(x)\right ) y(x)^3}-9 a y(x)^2}\right )-27 a y(x)^2 \left (3 \sqrt {3} \sqrt {-a^2 \left (a K[1]^3-27 y(x)\right ) y(x)^3}-27 a y(x)^2\right )^{2/3}}{3 a^2 K[1]^2 \left (a K[1]^3-27 y(x)\right ) y(x)^2}dK[1]+\int _1^{y(x)}\frac {3 a^3 K[2]^2 x^4-a^2 K[2] \left (3 \sqrt {3} \sqrt {-a^2 \left (a x^3-27 K[2]\right ) K[2]^3}-27 a K[2]^2\right )^{2/3} x^3+3^{5/6} a \sqrt {-a^2 \left (a x^3-27 K[2]\right ) K[2]^3} \sqrt [3]{\sqrt {3} \sqrt {-a^2 \left (a x^3-27 K[2]\right ) K[2]^3}-9 a K[2]^2} x^2-81 a^2 K[2]^3 x-9 a^2 \left (a x^3-27 K[2]\right ) K[2]^3 \int _1^x-\frac {a^3 K[2]^3 \left (-3^{5/6} a^2 K[2]^2 K[1]^3+\sqrt [3]{3} \sqrt {a^2 K[2]^3 \left (27 K[2]-a K[1]^3\right )} \left (\sqrt [3]{3 \sqrt {3} \sqrt {-a^2 \left (a K[1]^3-27 K[2]\right ) K[2]^3}-27 a K[2]^2} K[1]+18 K[2]\right )-27 \sqrt [6]{3} a K[2]^2 \left (\sqrt [3]{\sqrt {3} \sqrt {a^2 K[2]^3 \left (27 K[2]-a K[1]^3\right )}-9 a K[2]^2} K[1]+2\ 3^{2/3} K[2]\right )\right )}{2 \left (-a^2 \left (a K[1]^3-27 K[2]\right ) K[2]^3\right )^{3/2} \left (\sqrt {3} \sqrt {-a^2 \left (a K[1]^3-27 K[2]\right ) K[2]^3}-9 a K[2]^2\right )^{2/3}}dK[1] x+9 \sqrt [6]{3} \sqrt {-a^2 \left (a x^3-27 K[2]\right ) K[2]^3} \left (\sqrt {3} \sqrt {-a^2 \left (a x^3-27 K[2]\right ) K[2]^3}-9 a K[2]^2\right )^{2/3}+27 a K[2]^2 \left (3 \sqrt {3} \sqrt {-a^2 \left (a x^3-27 K[2]\right ) K[2]^3}-27 a K[2]^2\right )^{2/3}}{9 a^2 K[2]^3 \left (a x^4-27 x K[2]\right )}dK[2],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {\sqrt [3]{\sqrt {3} \sqrt {-a^2 \left (a K[3]^3-27 y(x)\right ) y(x)^3}-9 a y(x)^2} \left (-\left (\left (-3 i+\sqrt {3}\right ) a^2 y(x) \sqrt [3]{\sqrt {3} \sqrt {-a^2 \left (a K[3]^3-27 y(x)\right ) y(x)^3}-9 a y(x)^2} K[3]^3\right )+\sqrt {-a^2 \left (a K[3]^3-27 y(x)\right ) y(x)^3} \left (\sqrt [6]{3} \left (3 i+\sqrt {3}\right ) a K[3]^2+9 \left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {3} \sqrt {-a^2 \left (a K[3]^3-27 y(x)\right ) y(x)^3}-9 a y(x)^2}\right )+27 \left (-3 i+\sqrt {3}\right ) a y(x)^2 \sqrt [3]{\sqrt {3} \sqrt {-a^2 \left (a K[3]^3-27 y(x)\right ) y(x)^3}-9 a y(x)^2}\right )}{2\ 3^{5/6} a^2 K[3]^2 \left (a K[3]^3-27 y(x)\right ) y(x)^2}dK[3]+\int _1^{y(x)}\frac {6 a^3 K[4]^2 x^4+a^2 K[4] \left (\sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) x^2 \left (\sqrt {3} \sqrt {a^2 K[4]^3 \left (27 K[4]-a x^3\right )}-9 a K[4]^2\right )^{2/3}-162 K[4]^2\right ) x-18 a^2 \left (a x^3-27 K[4]\right ) K[4]^3 \int _1^x-\frac {\sqrt [6]{3} a^3 K[4]^3 \left (\sqrt [6]{3} \left (3 i+\sqrt {3}\right ) a^2 K[4]^2 K[3]^3+27 a K[4]^2 \left (\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {3} \sqrt {a^2 K[4]^3 \left (27 K[4]-a K[3]^3\right )}-9 a K[4]^2} K[3]+2 \sqrt [6]{3} \left (3 i+\sqrt {3}\right ) K[4]\right )-\sqrt {-a^2 \left (a K[3]^3-27 K[4]\right ) K[4]^3} \left (\left (-3 i+\sqrt {3}\right ) \sqrt [3]{\sqrt {3} \sqrt {a^2 K[4]^3 \left (27 K[4]-a K[3]^3\right )}-9 a K[4]^2} K[3]+18 \left (\sqrt [6]{3}+i 3^{2/3}\right ) K[4]\right )\right )}{4 \left (-a^2 \left (a K[3]^3-27 K[4]\right ) K[4]^3\right )^{3/2} \left (\sqrt {3} \sqrt {-a^2 \left (a K[3]^3-27 K[4]\right ) K[4]^3}-9 a K[4]^2\right )^{2/3}}dK[3] x-\sqrt [6]{3} a \sqrt [3]{\sqrt {3} \sqrt {-a^2 \left (a x^3-27 K[4]\right ) K[4]^3}-9 a K[4]^2} \left (\sqrt [6]{3} \left (3 i+\sqrt {3}\right ) \sqrt {-a^2 \left (a x^3-27 K[4]\right ) K[4]^3} x^2+27 \left (-3 i+\sqrt {3}\right ) K[4]^2 \sqrt [3]{\sqrt {3} \sqrt {a^2 K[4]^3 \left (27 K[4]-a x^3\right )}-9 a K[4]^2}\right )+9 i \sqrt [6]{3} \left (i+\sqrt {3}\right ) \sqrt {-a^2 \left (a x^3-27 K[4]\right ) K[4]^3} \left (\sqrt {3} \sqrt {-a^2 \left (a x^3-27 K[4]\right ) K[4]^3}-9 a K[4]^2\right )^{2/3}}{18 a^2 K[4]^3 \left (a x^4-27 x K[4]\right )}dK[4],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {\sqrt [3]{\sqrt {3} \sqrt {-a^2 \left (a K[5]^3-27 y(x)\right ) y(x)^3}-9 a y(x)^2} \left (-\left (\left (3 i+\sqrt {3}\right ) a^2 y(x) \sqrt [3]{\sqrt {3} \sqrt {-a^2 \left (a K[5]^3-27 y(x)\right ) y(x)^3}-9 a y(x)^2} K[5]^3\right )+\sqrt {-a^2 \left (a K[5]^3-27 y(x)\right ) y(x)^3} \left (\sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) a K[5]^2+9 \left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {3} \sqrt {-a^2 \left (a K[5]^3-27 y(x)\right ) y(x)^3}-9 a y(x)^2}\right )+27 \left (3 i+\sqrt {3}\right ) a y(x)^2 \sqrt [3]{\sqrt {3} \sqrt {-a^2 \left (a K[5]^3-27 y(x)\right ) y(x)^3}-9 a y(x)^2}\right )}{2\ 3^{5/6} a^2 K[5]^2 \left (a K[5]^3-27 y(x)\right ) y(x)^2}dK[5]+\int _1^{y(x)}-\frac {-6 a^3 K[6]^2 x^4-a^2 K[6] \left (\sqrt [6]{3} \left (3 i+\sqrt {3}\right ) x^2 \left (\sqrt {3} \sqrt {a^2 K[6]^3 \left (27 K[6]-a x^3\right )}-9 a K[6]^2\right )^{2/3}-162 K[6]^2\right ) x+18 a^2 \left (a x^3-27 K[6]\right ) K[6]^3 \int _1^x-\frac {\sqrt [6]{3} a^3 K[6]^3 \left (\sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) a^2 K[6]^2 K[5]^3+27 a K[6]^2 \left (\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {3} \sqrt {a^2 K[6]^3 \left (27 K[6]-a K[5]^3\right )}-9 a K[6]^2} K[5]+2 \sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) K[6]\right )+i \sqrt {a^2 K[6]^3 \left (27 K[6]-a K[5]^3\right )} \left (i \left (3 i+\sqrt {3}\right ) \sqrt [3]{\sqrt {3} \sqrt {a^2 K[6]^3 \left (27 K[6]-a K[5]^3\right )}-9 a K[6]^2} K[5]+18 \sqrt [6]{3} \left (i+\sqrt {3}\right ) K[6]\right )\right )}{4 \left (-a^2 \left (a K[5]^3-27 K[6]\right ) K[6]^3\right )^{3/2} \left (\sqrt {3} \sqrt {-a^2 \left (a K[5]^3-27 K[6]\right ) K[6]^3}-9 a K[6]^2\right )^{2/3}}dK[5] x+\sqrt [6]{3} a \sqrt [3]{\sqrt {3} \sqrt {-a^2 \left (a x^3-27 K[6]\right ) K[6]^3}-9 a K[6]^2} \left (\sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) \sqrt {-a^2 \left (a x^3-27 K[6]\right ) K[6]^3} x^2+27 \left (3 i+\sqrt {3}\right ) K[6]^2 \sqrt [3]{\sqrt {3} \sqrt {a^2 K[6]^3 \left (27 K[6]-a x^3\right )}-9 a K[6]^2}\right )+9 \sqrt [6]{3} \left (1+i \sqrt {3}\right ) \sqrt {-a^2 \left (a x^3-27 K[6]\right ) K[6]^3} \left (\sqrt {3} \sqrt {-a^2 \left (a x^3-27 K[6]\right ) K[6]^3}-9 a K[6]^2\right )^{2/3}}{18 a^2 K[6]^3 \left (a x^4-27 x K[6]\right )}dK[6],y(x)\right ]\right \}\]
Maple ✓
cpu = 3.417 (sec), leaf count = 39
\[\left [y \left (x \right ) = \frac {a \,x^{3}}{27}, y \left (x \right ) = \frac {x^{2}}{4 \textit {\_C1}}-\frac {x}{2 \textit {\_C1}^{2} a}+\frac {1}{4 \textit {\_C1}^{3} a^{2}}\right ]\] Mathematica raw input
DSolve[2*a*y[x]^2 - a*x*y[x]*y'[x] + y'[x]^3 == 0,y[x],x]
Mathematica raw output
{Solve[C[1] == Inactive[Integrate][(a^2*K[1]^3*y[x]*(-27*a*y[x]^2 + 3*Sqrt[3]*Sqrt[-(a^2*(a*K[1]^3 - 27*y[x])*y[x]^3)])^(2/3) - 27*a*y[x]^2*(-27*a*y[x]^2 + 3*Sqrt[3]*Sqrt[-(a^2*(a*K[1]^3 - 27*y[x])*y[x]^3)])^(2/3) - 3^(1/6)*Sqrt[-(a^2*(a*K[1]^3 - 27*y[x])*y[x]^3)]*(-9*a*y[x]^2 + Sqrt[3]*Sqrt[-(a^2*(a*K[1]^3 - 27*y[x])*y[x]^3)])^(1/3)*(3^(2/3)*a*K[1]^2 + 9*(-9*a*y[x]^2 + Sqrt[3]*Sqrt[-(a^2*(a*K[1]^3 - 27*y[x])*y[x]^3)])^(1/3)))/(3*a^2*K[1]^2*(a*K[1]^3 - 27*y[x])*y[x]^2), {K[1], 1, x}] + Inactive[Integrate][(3*a^3*x^4*K[2]^2 - 81*a^2*x*K[2]^3 + 3^(5/6)*a*x^2*Sqrt[-(a^2*(a*x^3 - 27*K[2])*K[2]^3)]*(-9*a*K[2]^2 + Sqrt[3]*Sqrt[-(a^2*(a*x^3 - 27*K[2])*K[2]^3)])^(1/3) + 9*3^(1/6)*Sqrt[-(a^2*(a*x^3 - 27*K[2])*K[2]^3)]*(-9*a*K[2]^2 + Sqrt[3]*Sqrt[-(a^2*(a*x^3 - 27*K[2])*K[2]^3)])^(2/3) - a^2*x^3*K[2]*(-27*a*K[2]^2 + 3*Sqrt[3]*Sqrt[-(a^2*(a*x^3 - 27*K[2])*K[2]^3)])^(2/3) + 27*a*K[2]^2*(-27*a*K[2]^2 + 3*Sqrt[3]*Sqrt[-(a^2*(a*x^3 - 27*K[2])*K[2]^3)])^(2/3) - 9*a^2*x*(a*x^3 - 27*K[2])*K[2]^3*Inactive[Integrate][-1/2*(a^3*K[2]^3*(-(3^(5/6)*a^2*K[1]^3*K[2]^2) + 3^(1/3)*Sqrt[a^2*K[2]^3*(-(a*K[1]^3) + 27*K[2])]*(18*K[2] + K[1]*(-27*a*K[2]^2 + 3*Sqrt[3]*Sqrt[-(a^2*(a*K[1]^3 - 27*K[2])*K[2]^3)])^(1/3)) - 27*3^(1/6)*a*K[2]^2*(2*3^(2/3)*K[2] + K[1]*(-9*a*K[2]^2 + Sqrt[3]*Sqrt[a^2*K[2]^3*(-(a*K[1]^3) + 27*K[2])])^(1/3))))/((-(a^2*(a*K[1]^3 - 27*K[2])*K[2]^3))^(3/2)*(-9*a*K[2]^2 + Sqrt[3]*Sqrt[-(a^2*(a*K[1]^3 - 27*K[2])*K[2]^3)])^(2/3)), {K[1], 1, x}])/(9*a^2*K[2]^3*(a*x^4 - 27*x*K[2])), {K[2], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][((-9*a*y[x]^2 + Sqrt[3]*Sqrt[-(a^2*(a*K[3]^3 - 27*y[x])*y[x]^3)])^(1/3)*(-((-3*I + Sqrt[3])*a^2*K[3]^3*y[x]*(-9*a*y[x]^2 + Sqrt[3]*Sqrt[-(a^2*(a*K[3]^3 - 27*y[x])*y[x]^3)])^(1/3)) + 27*(-3*I + Sqrt[3])*a*y[x]^2*(-9*a*y[x]^2 + Sqrt[3]*Sqrt[-(a^2*(a*K[3]^3 - 27*y[x])*y[x]^3)])^(1/3) + Sqrt[-(a^2*(a*K[3]^3 - 27*y[x])*y[x]^3)]*(3^(1/6)*(3*I + Sqrt[3])*a*K[3]^2 + 9*(1 - I*Sqrt[3])*(-9*a*y[x]^2 + Sqrt[3]*Sqrt[-(a^2*(a*K[3]^3 - 27*y[x])*y[x]^3)])^(1/3))))/(2*3^(5/6)*a^2*K[3]^2*(a*K[3]^3 - 27*y[x])*y[x]^2), {K[3], 1, x}] + Inactive[Integrate][(6*a^3*x^4*K[4]^2 + (9*I)*3^(1/6)*(I + Sqrt[3])*Sqrt[-(a^2*(a*x^3 - 27*K[4])*K[4]^3)]*(-9*a*K[4]^2 + Sqrt[3]*Sqrt[-(a^2*(a*x^3 - 27*K[4])*K[4]^3)])^(2/3) - 3^(1/6)*a*(-9*a*K[4]^2 + Sqrt[3]*Sqrt[-(a^2*(a*x^3 - 27*K[4])*K[4]^3)])^(1/3)*(3^(1/6)*(3*I + Sqrt[3])*x^2*Sqrt[-(a^2*(a*x^3 - 27*K[4])*K[4]^3)] + 27*(-3*I + Sqrt[3])*K[4]^2*(-9*a*K[4]^2 + Sqrt[3]*Sqrt[a^2*K[4]^3*(-(a*x^3) + 27*K[4])])^(1/3)) + a^2*x*K[4]*(-162*K[4]^2 + 3^(1/6)*(-3*I + Sqrt[3])*x^2*(-9*a*K[4]^2 + Sqrt[3]*Sqrt[a^2*K[4]^3*(-(a*x^3) + 27*K[4])])^(2/3)) - 18*a^2*x*(a*x^3 - 27*K[4])*K[4]^3*Inactive[Integrate][-1/4*(3^(1/6)*a^3*K[4]^3*(3^(1/6)*(3*I + Sqrt[3])*a^2*K[3]^3*K[4]^2 + 27*a*K[4]^2*(2*3^(1/6)*(3*I + Sqrt[3])*K[4] + (1 - I*Sqrt[3])*K[3]*(-9*a*K[4]^2 + Sqrt[3]*Sqrt[a^2*K[4]^3*(-(a*K[3]^3) + 27*K[4])])^(1/3)) - Sqrt[-(a^2*(a*K[3]^3 - 27*K[4])*K[4]^3)]*(18*(3^(1/6) + I*3^(2/3))*K[4] + (-3*I + Sqrt[3])*K[3]*(-9*a*K[4]^2 + Sqrt[3]*Sqrt[a^2*K[4]^3*(-(a*K[3]^3) + 27*K[4])])^(1/3))))/((-(a^2*(a*K[3]^3 - 27*K[4])*K[4]^3))^(3/2)*(-9*a*K[4]^2 + Sqrt[3]*Sqrt[-(a^2*(a*K[3]^3 - 27*K[4])*K[4]^3)])^(2/3)), {K[3], 1, x}])/(18*a^2*K[4]^3*(a*x^4 - 27*x*K[4])), {K[4], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][((-9*a*y[x]^2 + Sqrt[3]*Sqrt[-(a^2*(a*K[5]^3 - 27*y[x])*y[x]^3)])^(1/3)*(-((3*I + Sqrt[3])*a^2*K[5]^3*y[x]*(-9*a*y[x]^2 + Sqrt[3]*Sqrt[-(a^2*(a*K[5]^3 - 27*y[x])*y[x]^3)])^(1/3)) + 27*(3*I + Sqrt[3])*a*y[x]^2*(-9*a*y[x]^2 + Sqrt[3]*Sqrt[-(a^2*(a*K[5]^3 - 27*y[x])*y[x]^3)])^(1/3) + Sqrt[-(a^2*(a*K[5]^3 - 27*y[x])*y[x]^3)]*(3^(1/6)*(-3*I + Sqrt[3])*a*K[5]^2 + 9*(1 + I*Sqrt[3])*(-9*a*y[x]^2 + Sqrt[3]*Sqrt[-(a^2*(a*K[5]^3 - 27*y[x])*y[x]^3)])^(1/3))))/(2*3^(5/6)*a^2*K[5]^2*(a*K[5]^3 - 27*y[x])*y[x]^2), {K[5], 1, x}] + Inactive[Integrate][-1/18*(-6*a^3*x^4*K[6]^2 + 9*3^(1/6)*(1 + I*Sqrt[3])*Sqrt[-(a^2*(a*x^3 - 27*K[6])*K[6]^3)]*(-9*a*K[6]^2 + Sqrt[3]*Sqrt[-(a^2*(a*x^3 - 27*K[6])*K[6]^3)])^(2/3) + 3^(1/6)*a*(-9*a*K[6]^2 + Sqrt[3]*Sqrt[-(a^2*(a*x^3 - 27*K[6])*K[6]^3)])^(1/3)*(3^(1/6)*(-3*I + Sqrt[3])*x^2*Sqrt[-(a^2*(a*x^3 - 27*K[6])*K[6]^3)] + 27*(3*I + Sqrt[3])*K[6]^2*(-9*a*K[6]^2 + Sqrt[3]*Sqrt[a^2*K[6]^3*(-(a*x^3) + 27*K[6])])^(1/3)) - a^2*x*K[6]*(-162*K[6]^2 + 3^(1/6)*(3*I + Sqrt[3])*x^2*(-9*a*K[6]^2 + Sqrt[3]*Sqrt[a^2*K[6]^3*(-(a*x^3) + 27*K[6])])^(2/3)) + 18*a^2*x*(a*x^3 - 27*K[6])*K[6]^3*Inactive[Integrate][-1/4*(3^(1/6)*a^3*K[6]^3*(3^(1/6)*(-3*I + Sqrt[3])*a^2*K[5]^3*K[6]^2 + 27*a*K[6]^2*(2*3^(1/6)*(-3*I + Sqrt[3])*K[6] + (1 + I*Sqrt[3])*K[5]*(-9*a*K[6]^2 + Sqrt[3]*Sqrt[a^2*K[6]^3*(-(a*K[5]^3) + 27*K[6])])^(1/3)) + I*Sqrt[a^2*K[6]^3*(-(a*K[5]^3) + 27*K[6])]*(18*3^(1/6)*(I + Sqrt[3])*K[6] + I*(3*I + Sqrt[3])*K[5]*(-9*a*K[6]^2 + Sqrt[3]*Sqrt[a^2*K[6]^3*(-(a*K[5]^3) + 27*K[6])])^(1/3))))/((-(a^2*(a*K[5]^3 - 27*K[6])*K[6]^3))^(3/2)*(-9*a*K[6]^2 + Sqrt[3]*Sqrt[-(a^2*(a*K[5]^3 - 27*K[6])*K[6]^3)])^(2/3)), {K[5], 1, x}])/(a^2*K[6]^3*(a*x^4 - 27*x*K[6])), {K[6], 1, y[x]}], y[x]]}
Maple raw input
dsolve(diff(y(x),x)^3-a*x*y(x)*diff(y(x),x)+2*a*y(x)^2 = 0, y(x))
Maple raw output
[y(x) = 1/27*a*x^3, y(x) = 1/4/_C1*x^2-1/2/_C1^2/a*x+1/4/_C1^3/a^2]