ODE
\[ y'(x)^3-2 y(x) y'(x)+y(x)^2=0 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Independent variable missing, Solve for \(y\)
Mathematica ✓
cpu = 0.164134 (sec), leaf count = 422
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int \frac {\sqrt [3]{\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2}}{\sqrt [3]{2} \left (\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2\right )^{2/3}+4 \sqrt [3]{3} \text {$\#$1}}d\text {$\#$1}\& \right ]\left [\frac {x}{6^{2/3}}+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int \frac {\sqrt [3]{\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2}}{\sqrt [3]{2} 3^{2/3} \left (\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2\right )^{2/3}-\sqrt [3]{2} \sqrt [6]{3} i \left (\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2\right )^{2/3}-12 \text {$\#$1}-4 i \text {$\#$1} \sqrt {3}}d\text {$\#$1}\& \right ]\left [c_1-\frac {i x}{2\ 2^{2/3} 3^{5/6}}\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int \frac {\sqrt [3]{\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2}}{\sqrt [3]{2} 3^{2/3} \left (\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2\right )^{2/3}+\sqrt [3]{2} \sqrt [6]{3} i \left (\sqrt {3} \sqrt {\text {$\#$1}^3 (27 \text {$\#$1}-32)}-9 \text {$\#$1}^2\right )^{2/3}-12 \text {$\#$1}+4 i \text {$\#$1} \sqrt {3}}d\text {$\#$1}\& \right ]\left [\frac {i x}{2\ 2^{2/3} 3^{5/6}}+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 0.314 (sec), leaf count = 241
\[\left [x -\left (\int _{}^{y \left (x \right )}\frac {6 \left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {1}{3}}}{\left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {2}{3}}+24 \textit {\_a}}d \textit {\_a} \right )-\textit {\_C1} = 0, x -\left (\int _{}^{y \left (x \right )}\frac {12 \left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {1}{3}}}{\left (1+i \sqrt {3}\right ) \left (12 i \sqrt {3}\, \textit {\_a} -\left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {2}{3}}+12 \textit {\_a} \right )}d \textit {\_a} \right )-\textit {\_C1} = 0, x -\left (\int _{}^{y \left (x \right )}\frac {12 \left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {1}{3}}}{\left (-1+i \sqrt {3}\right ) \left (12 i \sqrt {3}\, \textit {\_a} +\left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {2}{3}}-12 \textit {\_a} \right )}d \textit {\_a} \right )-\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[y[x]^2 - 2*y[x]*y'[x] + y'[x]^3 == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[Inactive[Integrate][(-9*#1^2 + Sqrt[3]*Sqrt[#1^3*(-32 + 27*#1)])^(1/3)/(4*3^(1/3)*#1 + 2^(1/3)*(-9*#1^2 + Sqrt[3]*Sqrt[#1^3*(-32 + 27*#1)])^(2/3)), #1] & ][x/6^(2/3) + C[1]]}, {y[x] -> InverseFunction[Inactive[Integrate][(-9*#1^2 + Sqrt[3]*Sqrt[#1^3*(-32 + 27*#1)])^(1/3)/(-12*#1 - 4*I*Sqrt[3]*#1 - I*2^(1/3)*3^(1/6)*(-9*#1^2 + Sqrt[3]*Sqrt[#1^3*(-32 + 27*#1)])^(2/3) + 2^(1/3)*3^(2/3)*(-9*#1^2 + Sqrt[3]*Sqrt[#1^3*(-32 + 27*#1)])^(2/3)), #1] & ][((-1/2*I)*x)/(2^(2/3)*3^(5/6)) + C[1]]}, {y[x] -> InverseFunction[Inactive[Integrate][(-9*#1^2 + Sqrt[3]*Sqrt[#1^3*(-32 + 27*#1)])^(1/3)/(-12*#1 + 4*I*Sqrt[3]*#1 + I*2^(1/3)*3^(1/6)*(-9*#1^2 + Sqrt[3]*Sqrt[#1^3*(-32 + 27*#1)])^(2/3) + 2^(1/3)*3^(2/3)*(-9*#1^2 + Sqrt[3]*Sqrt[#1^3*(-32 + 27*#1)])^(2/3)), #1] & ][((I/2)*x)/(2^(2/3)*3^(5/6)) + C[1]]}}
Maple raw input
dsolve(diff(y(x),x)^3-2*y(x)*diff(y(x),x)+y(x)^2 = 0, y(x))
Maple raw output
[x-Intat(6/((-108*_a^2+12*(81*_a^4-96*_a^3)^(1/2))^(2/3)+24*_a)*(-108*_a^2+12*(81*_a^4-96*_a^3)^(1/2))^(1/3),_a = y(x))-_C1 = 0, x-Intat(12/(1+I*3^(1/2))/(12*I*3^(1/2)*_a-(-108*_a^2+12*(81*_a^4-96*_a^3)^(1/2))^(2/3)+12*_a)*(-108*_a^2+12*(81*_a^4-96*_a^3)^(1/2))^(1/3),_a = y(x))-_C1 = 0, x-Intat(12/(-1+I*3^(1/2))/(12*I*3^(1/2)*_a+(-108*_a^2+12*(81*_a^4-96*_a^3)^(1/2))^(2/3)-12*_a)*(-108*_a^2+12*(81*_a^4-96*_a^3)^(1/2))^(1/3),_a = y(x))-_C1 = 0]