Internal problem ID [4262]
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]
\[ \boxed {{y^{\prime }}^{3}+{y^{\prime }}^{2}-y=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 433
dsolve(diff(y(x),x)^3+diff(y(x),x)^2-y(x) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ 3 \,2^{\frac {1}{3}} \sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {\left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{\frac {1}{3}}}{\sqrt {3}\, 2^{\frac {1}{3}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{\frac {1}{3}}-3^{\frac {1}{3}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{\frac {2}{3}}-3^{\frac {2}{3}} 2^{\frac {2}{3}}}d \textit {\_a} \right )+x -c_{1} &= 0 \\ \frac {12 \,2^{\frac {1}{3}} \sqrt {3}\, \left (\int _{}^{y \left (x \right )}-\frac {\left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{\frac {1}{3}}}{\left (2^{\frac {1}{3}} 3^{\frac {1}{3}}+3^{\frac {1}{6}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{\frac {1}{3}}\right ) \left (i 3^{\frac {5}{6}} 2^{\frac {1}{3}}+2^{\frac {1}{3}} 3^{\frac {1}{3}}-2 \,3^{\frac {1}{6}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{\frac {1}{3}}\right )}d \textit {\_a} \right )+\left (x -c_{1} \right ) \left (1+i \sqrt {3}\right )}{1+i \sqrt {3}} &= 0 \\ \frac {12 i 2^{\frac {1}{3}} \sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {\left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{\frac {1}{3}}}{\left (2^{\frac {1}{3}} 3^{\frac {1}{3}}+3^{\frac {1}{6}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{\frac {1}{3}}\right ) \left (i 3^{\frac {5}{6}} 2^{\frac {1}{3}}+2 \,3^{\frac {1}{6}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{\frac {1}{3}}-2^{\frac {1}{3}} 3^{\frac {1}{3}}\right )}d \textit {\_a} \right )+\left (x -c_{1} \right ) \left (\sqrt {3}+i\right )}{\sqrt {3}+i} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 105.918 (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*}