ODE
\[ y'(x)^3-y(x) y'(x)^2+y(x)^2=0 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Independent variable missing, Solve for \(y\)
Mathematica ✓
cpu = 8.65166 (sec), leaf count = 648
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{2 K[1]^3-27 K[1]^2+3 \sqrt {3} \sqrt {-K[1]^4 (4 K[1]-27)}}}{2 \sqrt [3]{2} K[1]^2+2 \sqrt [3]{2 K[1]^3-27 K[1]^2+3 \sqrt {3} \sqrt {-K[1]^4 (4 K[1]-27)}} K[1]+2^{2/3} \left (2 K[1]^3-27 K[1]^2+3 \sqrt {3} \sqrt {-K[1]^4 (4 K[1]-27)}\right )^{2/3}}dK[1]\& \right ]\left [\frac {x}{6}+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{2 K[2]^3-27 K[2]^2+3 \sqrt {3} \sqrt {-K[2]^4 (4 K[2]-27)}}}{2 i \sqrt [3]{2} \sqrt {3} K[2]^2-2 \sqrt [3]{2} K[2]^2+4 \sqrt [3]{2 K[2]^3-27 K[2]^2+3 \sqrt {3} \sqrt {-K[2]^4 (4 K[2]-27)}} K[2]-i 2^{2/3} \sqrt {3} \left (2 K[2]^3-27 K[2]^2+3 \sqrt {3} \sqrt {-K[2]^4 (4 K[2]-27)}\right )^{2/3}-2^{2/3} \left (2 K[2]^3-27 K[2]^2+3 \sqrt {3} \sqrt {-K[2]^4 (4 K[2]-27)}\right )^{2/3}}dK[2]\& \right ]\left [\frac {x}{12}+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{2 K[3]^3-27 K[3]^2+3 \sqrt {3} \sqrt {-K[3]^4 (4 K[3]-27)}}}{-2 i \sqrt [3]{2} \sqrt {3} K[3]^2-2 \sqrt [3]{2} K[3]^2+4 \sqrt [3]{2 K[3]^3-27 K[3]^2+3 \sqrt {3} \sqrt {-K[3]^4 (4 K[3]-27)}} K[3]+i 2^{2/3} \sqrt {3} \left (2 K[3]^3-27 K[3]^2+3 \sqrt {3} \sqrt {-K[3]^4 (4 K[3]-27)}\right )^{2/3}-2^{2/3} \left (2 K[3]^3-27 K[3]^2+3 \sqrt {3} \sqrt {-K[3]^4 (4 K[3]-27)}\right )^{2/3}}dK[3]\& \right ]\left [\frac {x}{12}+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 0.255 (sec), leaf count = 423
\[\left [y \left (x \right ) = 0, x -\left (\int _{}^{y \left (x \right )}\frac {6 \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{4 \textit {\_a}^{2}+2 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-3 \textit {\_a}^{4} \left (-27+4 \textit {\_a} \right )}\right )^{\frac {1}{3}}+\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}}d \textit {\_a} \right )-\textit {\_C1} = 0, x -\left (\int _{}^{y \left (x \right )}-\frac {12 \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, \textit {\_a}^{2}-4 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-3 \textit {\_a}^{4} \left (-27+4 \textit {\_a} \right )}\right )^{\frac {1}{3}}+\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}+4 \textit {\_a}^{2}}d \textit {\_a} \right )-\textit {\_C1} = 0, x -\left (\int _{}^{y \left (x \right )}\frac {12 \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, \textit {\_a}^{2}+4 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-3 \textit {\_a}^{4} \left (-27+4 \textit {\_a} \right )}\right )^{\frac {1}{3}}-\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}-4 \textit {\_a}^{2}}d \textit {\_a} \right )-\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[y[x]^2 - y[x]*y'[x]^2 + y'[x]^3 == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[Inactive[Integrate][(-27*K[1]^2 + 2*K[1]^3 + 3*Sqrt[3]*Sqrt[-(K[1]^4*(-27 + 4*K[1]))])^(1/3)/(2*2^(1/3)*K[1]^2 + 2*K[1]*(-27*K[1]^2 + 2*K[1]^3 + 3*Sqrt[3]*Sqrt[-(K[1]^4*(-27 + 4*K[1]))])^(1/3) + 2^(2/3)*(-27*K[1]^2 + 2*K[1]^3 + 3*Sqrt[3]*Sqrt[-(K[1]^4*(-27 + 4*K[1]))])^(2/3)), {K[1], 1, #1}] & ][x/6 + C[1]]}, {y[x] -> InverseFunction[Inactive[Integrate][(-27*K[2]^2 + 2*K[2]^3 + 3*Sqrt[3]*Sqrt[-(K[2]^4*(-27 + 4*K[2]))])^(1/3)/(-2*2^(1/3)*K[2]^2 + (2*I)*2^(1/3)*Sqrt[3]*K[2]^2 + 4*K[2]*(-27*K[2]^2 + 2*K[2]^3 + 3*Sqrt[3]*Sqrt[-(K[2]^4*(-27 + 4*K[2]))])^(1/3) - 2^(2/3)*(-27*K[2]^2 + 2*K[2]^3 + 3*Sqrt[3]*Sqrt[-(K[2]^4*(-27 + 4*K[2]))])^(2/3) - I*2^(2/3)*Sqrt[3]*(-27*K[2]^2 + 2*K[2]^3 + 3*Sqrt[3]*Sqrt[-(K[2]^4*(-27 + 4*K[2]))])^(2/3)), {K[2], 1, #1}] & ][x/12 + C[1]]}, {y[x] -> InverseFunction[Inactive[Integrate][(-27*K[3]^2 + 2*K[3]^3 + 3*Sqrt[3]*Sqrt[-(K[3]^4*(-27 + 4*K[3]))])^(1/3)/(-2*2^(1/3)*K[3]^2 - (2*I)*2^(1/3)*Sqrt[3]*K[3]^2 + 4*K[3]*(-27*K[3]^2 + 2*K[3]^3 + 3*Sqrt[3]*Sqrt[-(K[3]^4*(-27 + 4*K[3]))])^(1/3) - 2^(2/3)*(-27*K[3]^2 + 2*K[3]^3 + 3*Sqrt[3]*Sqrt[-(K[3]^4*(-27 + 4*K[3]))])^(2/3) + I*2^(2/3)*Sqrt[3]*(-27*K[3]^2 + 2*K[3]^3 + 3*Sqrt[3]*Sqrt[-(K[3]^4*(-27 + 4*K[3]))])^(2/3)), {K[3], 1, #1}] & ][x/12 + C[1]]}}
Maple raw input
dsolve(diff(y(x),x)^3-y(x)*diff(y(x),x)^2+y(x)^2 = 0, y(x))
Maple raw output
[y(x) = 0, x-Intat(6/(4*_a^2+2*_a*(-108*_a^2+8*_a^3+12*(-3*_a^4*(-27+4*_a))^(1/2))^(1/3)+(-108*_a^2+8*_a^3+12*(-12*_a^5+81*_a^4)^(1/2))^(2/3))*(-108*_a^2+8*_a^3+12*(-12*_a^5+81*_a^4)^(1/2))^(1/3),_a = y(x))-_C1 = 0, x-Intat(-12/(I*3^(1/2)*(-108*_a^2+8*_a^3+12*(-12*_a^5+81*_a^4)^(1/2))^(2/3)-4*I*3^(1/2)*_a^2-4*_a*(-108*_a^2+8*_a^3+12*(-3*_a^4*(-27+4*_a))^(1/2))^(1/3)+(-108*_a^2+8*_a^3+12*(-12*_a^5+81*_a^4)^(1/2))^(2/3)+4*_a^2)*(-108*_a^2+8*_a^3+12*(-12*_a^5+81*_a^4)^(1/2))^(1/3),_a = y(x))-_C1 = 0, x-Intat(12/(I*3^(1/2)*(-108*_a^2+8*_a^3+12*(-12*_a^5+81*_a^4)^(1/2))^(2/3)-4*I*3^(1/2)*_a^2+4*_a*(-108*_a^2+8*_a^3+12*(-3*_a^4*(-27+4*_a))^(1/2))^(1/3)-(-108*_a^2+8*_a^3+12*(-12*_a^5+81*_a^4)^(1/2))^(2/3)-4*_a^2)*(-108*_a^2+8*_a^3+12*(-12*_a^5+81*_a^4)^(1/2))^(1/3),_a = y(x))-_C1 = 0]