ODE
\[ 2 y'(x)^3+y'(x)^2-y(x)=0 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Independent variable missing, Solve for \(y\)
Mathematica ✓
cpu = 0.188313 (sec), leaf count = 1115
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int \frac {\sqrt [3]{-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2+\sqrt {4 \left (-1^2+3\ 2\ 0\right )^3+\left (-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2-0\ 2\ 9\right )^2}-0\ 2\ 9}}{2 \sqrt [3]{-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2+\sqrt {4 \left (-1^2+3\ 2\ 0\right )^3+\left (-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2-0\ 2\ 9\right )^2}-0\ 2\ 9}+2^{2/3} \left (-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2+\sqrt {4 \left (-1^2+3\ 2\ 0\right )^3+\left (-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2-0\ 2\ 9\right )^2}-0\ 2\ 9\right )^{2/3}+2\ 1^2 \sqrt [3]{2}-0\ 2\ 6 \sqrt [3]{2}}d\text {$\#$1}\& \right ]\left [c_1-\frac {x}{12}\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int \frac {\sqrt [3]{-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2+\sqrt {4 \left (-1^2+3\ 2\ 0\right )^3+\left (-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2-0\ 2\ 9\right )^2}-0\ 2\ 9}}{-4 \sqrt [3]{-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2+\sqrt {4 \left (-1^2+3\ 2\ 0\right )^3+\left (-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2-0\ 2\ 9\right )^2}-0\ 2\ 9}+2^{2/3} \left (-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2+\sqrt {4 \left (-1^2+3\ 2\ 0\right )^3+\left (-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2-0\ 2\ 9\right )^2}-0\ 2\ 9\right )^{2/3}-2^{2/3} i \sqrt {3} \left (-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2+\sqrt {4 \left (-1^2+3\ 2\ 0\right )^3+\left (-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2-0\ 2\ 9\right )^2}-0\ 2\ 9\right )^{2/3}+2\ 1^2 \sqrt [3]{2}-0\ 2\ 6 \sqrt [3]{2}+2\ 1^2 \sqrt [3]{2} i \sqrt {3}-0\ 2\ 6 \sqrt [3]{2} i \sqrt {3}}d\text {$\#$1}\& \right ]\left [\frac {x}{24}+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int \frac {\sqrt [3]{-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2+\sqrt {4 \left (-1^2+3\ 2\ 0\right )^3+\left (-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2-0\ 2\ 9\right )^2}-0\ 2\ 9}}{-4 \sqrt [3]{-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2+\sqrt {4 \left (-1^2+3\ 2\ 0\right )^3+\left (-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2-0\ 2\ 9\right )^2}-0\ 2\ 9}+2^{2/3} \left (-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2+\sqrt {4 \left (-1^2+3\ 2\ 0\right )^3+\left (-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2-0\ 2\ 9\right )^2}-0\ 2\ 9\right )^{2/3}+2^{2/3} i \sqrt {3} \left (-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2+\sqrt {4 \left (-1^2+3\ 2\ 0\right )^3+\left (-27 2^2 \text {$\#$1}+2\ 1^3-0\ 27\ 2^2-0\ 2\ 9\right )^2}-0\ 2\ 9\right )^{2/3}+2\ 1^2 \sqrt [3]{2}-0\ 2\ 6 \sqrt [3]{2}-2\ 1^2 \sqrt [3]{2} i \sqrt {3}+0\ 2\ 6 \sqrt [3]{2} i \sqrt {3}}d\text {$\#$1}\& \right ]\left [\frac {x}{24}+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 34.642 (sec), leaf count = 372
\[\left [y \left (x \right ) = 0, x -\left (\int _{}^{y \left (x \right )}\frac {6 \,3^{\frac {1}{3}} \left (-1+54 \textit {\_a} +6 \sqrt {81 \textit {\_a}^{2}-3 \textit {\_a}}\right )^{\frac {1}{3}}}{3^{\frac {1}{3}} \left (-1+54 \textit {\_a} +6 \sqrt {81 \textit {\_a}^{2}-3 \textit {\_a}}\right )^{\frac {2}{3}}+3^{\frac {1}{3}}-\left (\sqrt {3}\, \left (54 \sqrt {3}\, \textit {\_a} -\sqrt {3}+18 \sqrt {\textit {\_a} \left (-1+27 \textit {\_a} \right )}\right )\right )^{\frac {1}{3}}}d \textit {\_a} \right )-\textit {\_C1} = 0, x -\left (\int _{}^{y \left (x \right )}-\frac {12 \,3^{\frac {1}{3}} \left (-1+54 \textit {\_a} +6 \sqrt {81 \textit {\_a}^{2}-3 \textit {\_a}}\right )^{\frac {1}{3}}}{i 3^{\frac {5}{6}} \left (-1+54 \textit {\_a} +6 \sqrt {81 \textit {\_a}^{2}-3 \textit {\_a}}\right )^{\frac {2}{3}}-i 3^{\frac {5}{6}}+3^{\frac {1}{3}} \left (-1+54 \textit {\_a} +6 \sqrt {81 \textit {\_a}^{2}-3 \textit {\_a}}\right )^{\frac {2}{3}}+3^{\frac {1}{3}}+2 \left (\sqrt {3}\, \left (54 \sqrt {3}\, \textit {\_a} -\sqrt {3}+18 \sqrt {\textit {\_a} \left (-1+27 \textit {\_a} \right )}\right )\right )^{\frac {1}{3}}}d \textit {\_a} \right )-\textit {\_C1} = 0, x -\left (\int _{}^{y \left (x \right )}\frac {12 \,3^{\frac {1}{3}} \left (-1+54 \textit {\_a} +6 \sqrt {81 \textit {\_a}^{2}-3 \textit {\_a}}\right )^{\frac {1}{3}}}{i 3^{\frac {5}{6}} \left (-1+54 \textit {\_a} +6 \sqrt {81 \textit {\_a}^{2}-3 \textit {\_a}}\right )^{\frac {2}{3}}-i 3^{\frac {5}{6}}-3^{\frac {1}{3}} \left (-1+54 \textit {\_a} +6 \sqrt {81 \textit {\_a}^{2}-3 \textit {\_a}}\right )^{\frac {2}{3}}-3^{\frac {1}{3}}-2 \left (\sqrt {3}\, \left (54 \sqrt {3}\, \textit {\_a} -\sqrt {3}+18 \sqrt {\textit {\_a} \left (-1+27 \textit {\_a} \right )}\right )\right )^{\frac {1}{3}}}d \textit {\_a} \right )-\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[-y[x] + y'[x]^2 + 2*y'[x]^3 == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[Inactive[Integrate][(2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1 + Sqrt[4*(-1^2 + 3*2*0)^3 + (2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1)^2])^(1/3)/(2*2^(1/3)*1^2 - 6*2^(1/3)*2*0 + 2*1*(2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1 + Sqrt[4*(-1^2 + 3*2*0)^3 + (2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1)^2])^(1/3) + 2^(2/3)*(2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1 + Sqrt[4*(-1^2 + 3*2*0)^3 + (2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1)^2])^(2/3)), #1] & ][-1/12*x + C[1]]}, {y[x] -> InverseFunction[Inactive[Integrate][(2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1 + Sqrt[4*(-1^2 + 3*2*0)^3 + (2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1)^2])^(1/3)/(2*2^(1/3)*1^2 + 2*I*2^(1/3)*Sqrt[3]*1^2 - 6*2^(1/3)*2*0 - 6*I*2^(1/3)*Sqrt[3]*2*0 - 4*1*(2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1 + Sqrt[4*(-1^2 + 3*2*0)^3 + (2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1)^2])^(1/3) + 2^(2/3)*(2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1 + Sqrt[4*(-1^2 + 3*2*0)^3 + (2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1)^2])^(2/3) - I*2^(2/3)*Sqrt[3]*(2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1 + Sqrt[4*(-1^2 + 3*2*0)^3 + (2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1)^2])^(2/3)), #1] & ][x/24 + C[1]]}, {y[x] -> InverseFunction[Inactive[Integrate][(2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1 + Sqrt[4*(-1^2 + 3*2*0)^3 + (2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1)^2])^(1/3)/(2*2^(1/3)*1^2 - 2*I*2^(1/3)*Sqrt[3]*1^2 - 6*2^(1/3)*2*0 + 6*I*2^(1/3)*Sqrt[3]*2*0 - 4*1*(2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1 + Sqrt[4*(-1^2 + 3*2*0)^3 + (2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1)^2])^(1/3) + 2^(2/3)*(2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1 + Sqrt[4*(-1^2 + 3*2*0)^3 + (2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1)^2])^(2/3) + I*2^(2/3)*Sqrt[3]*(2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1 + Sqrt[4*(-1^2 + 3*2*0)^3 + (2*1^3 - 9*2*1*0 - 27*2^2*0 - 27*2^2*#1)^2])^(2/3)), #1] & ][x/24 + C[1]]}}
Maple raw input
dsolve(2*diff(y(x),x)^3+diff(y(x),x)^2-y(x) = 0, y(x))
Maple raw output
[y(x) = 0, x-Intat(6*3^(1/3)/(3^(1/3)*(-1+54*_a+6*(81*_a^2-3*_a)^(1/2))^(2/3)+3^(1/3)-(3^(1/2)*(54*3^(1/2)*_a-3^(1/2)+18*(_a*(-1+27*_a))^(1/2)))^(1/3))*(-1+54*_a+6*(81*_a^2-3*_a)^(1/2))^(1/3),_a = y(x))-_C1 = 0, x-Intat(-12*3^(1/3)/(I*3^(5/6)*(-1+54*_a+6*(81*_a^2-3*_a)^(1/2))^(2/3)-I*3^(5/6)+3^(1/3)*(-1+54*_a+6*(81*_a^2-3*_a)^(1/2))^(2/3)+3^(1/3)+2*(3^(1/2)*(54*3^(1/2)*_a-3^(1/2)+18*(_a*(-1+27*_a))^(1/2)))^(1/3))*(-1+54*_a+6*(81*_a^2-3*_a)^(1/2))^(1/3),_a = y(x))-_C1 = 0, x-Intat(12*3^(1/3)/(I*3^(5/6)*(-1+54*_a+6*(81*_a^2-3*_a)^(1/2))^(2/3)-I*3^(5/6)-3^(1/3)*(-1+54*_a+6*(81*_a^2-3*_a)^(1/2))^(2/3)-3^(1/3)-2*(3^(1/2)*(54*3^(1/2)*_a-3^(1/2)+18*(_a*(-1+27*_a))^(1/2)))^(1/3))*(-1+54*_a+6*(81*_a^2-3*_a)^(1/2))^(1/3),_a = y(x))-_C1 = 0]