4.21.47 \(y'(x)^3-y'(x)^2+y(x)^2=0\)

ODE
\[ y'(x)^3-y'(x)^2+y(x)^2=0 \] ODE Classification

[_quadrature]

Book solution method
Missing Variables ODE, Independent variable missing, Use new variable

Mathematica ✓
cpu = 5.86986 (sec), leaf count = 578

\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{-27 K[1]^2+3 \sqrt {3} \sqrt {K[1]^2 \left (27 K[1]^2-4\right )}+2}}{2^{2/3} \left (-27 K[1]^2+3 \sqrt {3} \sqrt {K[1]^2 \left (27 K[1]^2-4\right )}+2\right )^{2/3}+2 \sqrt [3]{-27 K[1]^2+3 \sqrt {3} \sqrt {K[1]^2 \left (27 K[1]^2-4\right )}+2}+2 \sqrt [3]{2}}dK[1]\& \right ]\left [\frac {x}{6}+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{-27 K[2]^2+3 \sqrt {3} \sqrt {K[2]^2 \left (27 K[2]^2-4\right )}+2}}{-i 2^{2/3} \sqrt {3} \left (-27 K[2]^2+3 \sqrt {3} \sqrt {K[2]^2 \left (27 K[2]^2-4\right )}+2\right )^{2/3}-2^{2/3} \left (-27 K[2]^2+3 \sqrt {3} \sqrt {K[2]^2 \left (27 K[2]^2-4\right )}+2\right )^{2/3}+4 \sqrt [3]{-27 K[2]^2+3 \sqrt {3} \sqrt {K[2]^2 \left (27 K[2]^2-4\right )}+2}+2 i \sqrt [3]{2} \sqrt {3}-2 \sqrt [3]{2}}dK[2]\& \right ]\left [\frac {x}{12}+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{-27 K[3]^2+3 \sqrt {3} \sqrt {K[3]^2 \left (27 K[3]^2-4\right )}+2}}{i 2^{2/3} \sqrt {3} \left (-27 K[3]^2+3 \sqrt {3} \sqrt {K[3]^2 \left (27 K[3]^2-4\right )}+2\right )^{2/3}-2^{2/3} \left (-27 K[3]^2+3 \sqrt {3} \sqrt {K[3]^2 \left (27 K[3]^2-4\right )}+2\right )^{2/3}+4 \sqrt [3]{-27 K[3]^2+3 \sqrt {3} \sqrt {K[3]^2 \left (27 K[3]^2-4\right )}+2}-2 i \sqrt [3]{2} \sqrt {3}-2 \sqrt [3]{2}}dK[3]\& \right ]\left [\frac {x}{12}+c_1\right ]\right \}\right \}\]

Maple ✓
cpu = 0.547 (sec), leaf count = 398

\[\left [x -\left (\int _{}^{y \left (x \right )}\frac {6 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{\left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}}+2 \left (-\frac {4 \sqrt {3}\, \left (27 \sqrt {3}\, \textit {\_a}^{2}-2 \sqrt {3}-9 \sqrt {\textit {\_a}^{2} \left (27 \textit {\_a}^{2}-4\right )}\right )}{3}\right )^{\frac {1}{3}}+4}d \textit {\_a} \right )-\textit {\_C1} = 0, x -\left (\int _{}^{y \left (x \right )}-\frac {12 i \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{i \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}}-\sqrt {3}\, \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}}-4 i \left (-\frac {4 \sqrt {3}\, \left (27 \sqrt {3}\, \textit {\_a}^{2}-2 \sqrt {3}-9 \sqrt {\textit {\_a}^{2} \left (27 \textit {\_a}^{2}-4\right )}\right )}{3}\right )^{\frac {1}{3}}+4 i+4 \sqrt {3}}d \textit {\_a} \right )-\textit {\_C1} = 0, x -\left (\int _{}^{y \left (x \right )}-\frac {12 i \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{i \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}}+\sqrt {3}\, \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}}-4 i \left (-\frac {4 \sqrt {3}\, \left (27 \sqrt {3}\, \textit {\_a}^{2}-2 \sqrt {3}-9 \sqrt {\textit {\_a}^{2} \left (27 \textit {\_a}^{2}-4\right )}\right )}{3}\right )^{\frac {1}{3}}+4 i-4 \sqrt {3}}d \textit {\_a} \right )-\textit {\_C1} = 0\right ]\] Mathematica raw input

DSolve[y[x]^2 - y'[x]^2 + y'[x]^3 == 0,y[x],x]

Mathematica raw output

{{y[x] -> InverseFunction[Inactive[Integrate][(2 - 27*K[1]^2 + 3*Sqrt[3]*Sqrt[K[
1]^2*(-4 + 27*K[1]^2)])^(1/3)/(2*2^(1/3) + 2*(2 - 27*K[1]^2 + 3*Sqrt[3]*Sqrt[K[1
]^2*(-4 + 27*K[1]^2)])^(1/3) + 2^(2/3)*(2 - 27*K[1]^2 + 3*Sqrt[3]*Sqrt[K[1]^2*(-
4 + 27*K[1]^2)])^(2/3)), {K[1], 1, #1}] & ][x/6 + C[1]]}, {y[x] -> InverseFuncti
on[Inactive[Integrate][(2 - 27*K[2]^2 + 3*Sqrt[3]*Sqrt[K[2]^2*(-4 + 27*K[2]^2)])
^(1/3)/(-2*2^(1/3) + (2*I)*2^(1/3)*Sqrt[3] + 4*(2 - 27*K[2]^2 + 3*Sqrt[3]*Sqrt[K
[2]^2*(-4 + 27*K[2]^2)])^(1/3) - 2^(2/3)*(2 - 27*K[2]^2 + 3*Sqrt[3]*Sqrt[K[2]^2*
(-4 + 27*K[2]^2)])^(2/3) - I*2^(2/3)*Sqrt[3]*(2 - 27*K[2]^2 + 3*Sqrt[3]*Sqrt[K[2
]^2*(-4 + 27*K[2]^2)])^(2/3)), {K[2], 1, #1}] & ][x/12 + C[1]]}, {y[x] -> Invers
eFunction[Inactive[Integrate][(2 - 27*K[3]^2 + 3*Sqrt[3]*Sqrt[K[3]^2*(-4 + 27*K[
3]^2)])^(1/3)/(-2*2^(1/3) - (2*I)*2^(1/3)*Sqrt[3] + 4*(2 - 27*K[3]^2 + 3*Sqrt[3]
*Sqrt[K[3]^2*(-4 + 27*K[3]^2)])^(1/3) - 2^(2/3)*(2 - 27*K[3]^2 + 3*Sqrt[3]*Sqrt[
K[3]^2*(-4 + 27*K[3]^2)])^(2/3) + I*2^(2/3)*Sqrt[3]*(2 - 27*K[3]^2 + 3*Sqrt[3]*S
qrt[K[3]^2*(-4 + 27*K[3]^2)])^(2/3)), {K[3], 1, #1}] & ][x/12 + C[1]]}}

Maple raw input

dsolve(diff(y(x),x)^3-diff(y(x),x)^2+y(x)^2 = 0, y(x))

Maple raw output

[x-Intat(6/((8-108*_a^2+12*(81*_a^4-12*_a^2)^(1/2))^(2/3)+2*(-4/3*3^(1/2)*(27*3^
(1/2)*_a^2-2*3^(1/2)-9*(_a^2*(27*_a^2-4))^(1/2)))^(1/3)+4)*(8-108*_a^2+12*(81*_a
^4-12*_a^2)^(1/2))^(1/3),_a = y(x))-_C1 = 0, x-Intat(-12*I/(I*(8-108*_a^2+12*(81
*_a^4-12*_a^2)^(1/2))^(2/3)-3^(1/2)*(8-108*_a^2+12*(81*_a^4-12*_a^2)^(1/2))^(2/3
)-4*I*(-4/3*3^(1/2)*(27*3^(1/2)*_a^2-2*3^(1/2)-9*(_a^2*(27*_a^2-4))^(1/2)))^(1/3
)+4*I+4*3^(1/2))*(8-108*_a^2+12*(81*_a^4-12*_a^2)^(1/2))^(1/3),_a = y(x))-_C1 = 
0, x-Intat(-12*I/(I*(8-108*_a^2+12*(81*_a^4-12*_a^2)^(1/2))^(2/3)+3^(1/2)*(8-108
*_a^2+12*(81*_a^4-12*_a^2)^(1/2))^(2/3)-4*I*(-4/3*3^(1/2)*(27*3^(1/2)*_a^2-2*3^(
1/2)-9*(_a^2*(27*_a^2-4))^(1/2)))^(1/3)+4*I-4*3^(1/2))*(8-108*_a^2+12*(81*_a^4-1
2*_a^2)^(1/2))^(1/3),_a = y(x))-_C1 = 0]