Internal problem ID [3754]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 35
Problem number: 1040.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{3}+\left (y^{\prime }\right )^{2}-y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 388
dsolve(diff(y(x),x)^3+diff(y(x),x)^2-y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ x -\left (\int _{}^{y \relax (x )}\frac {6 \,6^{\frac {1}{3}} \left (-8+108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}-12 \textit {\_a}}\right )^{\frac {1}{3}}}{6^{\frac {1}{3}} \left (-8+108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}-12 \textit {\_a}}\right )^{\frac {2}{3}}+4 \,6^{\frac {1}{3}}-4 \left (\sqrt {3}\, \left (27 \sqrt {3}\, \textit {\_a} -2 \sqrt {3}+9 \sqrt {\textit {\_a} \left (-4+27 \textit {\_a} \right )}\right )\right )^{\frac {1}{3}}}d \textit {\_a} \right )-c_{1} = 0 \\ x -\left (\int _{}^{y \relax (x )}-\frac {12 \,6^{\frac {1}{3}} \left (-8+108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}-12 \textit {\_a}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, 6^{\frac {1}{3}} \left (-8+108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}-12 \textit {\_a}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, 6^{\frac {1}{3}}+6^{\frac {1}{3}} \left (-8+108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}-12 \textit {\_a}}\right )^{\frac {2}{3}}+4 \,6^{\frac {1}{3}}+8 \left (\sqrt {3}\, \left (27 \sqrt {3}\, \textit {\_a} -2 \sqrt {3}+9 \sqrt {\textit {\_a} \left (-4+27 \textit {\_a} \right )}\right )\right )^{\frac {1}{3}}}d \textit {\_a} \right )-c_{1} = 0 \\ x -\left (\int _{}^{y \relax (x )}\frac {12 \,6^{\frac {1}{3}} \left (-8+108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}-12 \textit {\_a}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, 6^{\frac {1}{3}} \left (-8+108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}-12 \textit {\_a}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, 6^{\frac {1}{3}}-6^{\frac {1}{3}} \left (-8+108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}-12 \textit {\_a}}\right )^{\frac {2}{3}}-4 \,6^{\frac {1}{3}}-8 \left (\sqrt {3}\, \left (27 \sqrt {3}\, \textit {\_a} -2 \sqrt {3}+9 \sqrt {\textit {\_a} \left (-4+27 \textit {\_a} \right )}\right )\right )^{\frac {1}{3}}}d \textit {\_a} \right )-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 87.697 (sec). Leaf size: 515
DSolve[(y'[x])^3 + (y'[x])^2 -y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{-27 K[1]+3 \sqrt {3} \sqrt {K[1] (27 K[1]-4)}+2}}{2^{2/3} \left (-27 K[1]+3 \sqrt {3} \sqrt {K[1] (27 K[1]-4)}+2\right )^{2/3}+2 \sqrt [3]{-27 K[1]+3 \sqrt {3} \sqrt {K[1] (27 K[1]-4)}+2}+2 \sqrt [3]{2}}dK[1]\&\right ]\left [-\frac {x}{6}+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{-27 K[2]+3 \sqrt {3} \sqrt {K[2] (27 K[2]-4)}+2}}{-i 2^{2/3} \sqrt {3} \left (-27 K[2]+3 \sqrt {3} \sqrt {K[2] (27 K[2]-4)}+2\right )^{2/3}+2^{2/3} \left (-27 K[2]+3 \sqrt {3} \sqrt {K[2] (27 K[2]-4)}+2\right )^{2/3}-4 \sqrt [3]{-27 K[2]+3 \sqrt {3} \sqrt {K[2] (27 K[2]-4)}+2}+2 i \sqrt [3]{2} \sqrt {3}+2 \sqrt [3]{2}}dK[2]\&\right ]\left [\frac {x}{12}+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{-27 K[3]+3 \sqrt {3} \sqrt {K[3] (27 K[3]-4)}+2}}{i 2^{2/3} \sqrt {3} \left (-27 K[3]+3 \sqrt {3} \sqrt {K[3] (27 K[3]-4)}+2\right )^{2/3}+2^{2/3} \left (-27 K[3]+3 \sqrt {3} \sqrt {K[3] (27 K[3]-4)}+2\right )^{2/3}-4 \sqrt [3]{-27 K[3]+3 \sqrt {3} \sqrt {K[3] (27 K[3]-4)}+2}-2 i \sqrt [3]{2} \sqrt {3}+2 \sqrt [3]{2}}dK[3]\&\right ]\left [\frac {x}{12}+c_1\right ] \\ y(x)\to 0 \\ \end{align*}