Internal problem ID [3755]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 35
Problem number: 1041.
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^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.218 (sec). Leaf size: 401
dsolve(diff(y(x),x)^3-diff(y(x),x)^2+y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ x -\left (\int _{}^{y \relax (x )}\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 )-c_{1} = 0 \\ x -\left (\int _{}^{y \relax (x )}\frac {12 i \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{\sqrt {3}\, \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}}-4 \sqrt {3}-i \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}d \textit {\_a} \right )-c_{1} = 0 \\ x -\left (\int _{}^{y \relax (x )}-\frac {12 i \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{\sqrt {3}\, \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}}+i \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 \sqrt {3}+4 i}d \textit {\_a} \right )-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 36.758 (sec). Leaf size: 583
DSolve[(y'[x])^3 - (y'[x])^2 +y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{-27 K[1]^2+3 \sqrt {3} \sqrt {K[1]^2 \left (27 K[1]^2-4\right )}+2}}{2^{2/3} \left (-27 K[1]^2+3 \sqrt {3} \sqrt {K[1]^2 \left (27 K[1]^2-4\right )}+2\right )^{2/3}+2 \sqrt [3]{-27 K[1]^2+3 \sqrt {3} \sqrt {K[1]^2 \left (27 K[1]^2-4\right )}+2}+2 \sqrt [3]{2}}dK[1]\&\right ]\left [\frac {x}{6}+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{-27 K[2]^2+3 \sqrt {3} \sqrt {K[2]^2 \left (27 K[2]^2-4\right )}+2}}{-i 2^{2/3} \sqrt {3} \left (-27 K[2]^2+3 \sqrt {3} \sqrt {K[2]^2 \left (27 K[2]^2-4\right )}+2\right )^{2/3}-2^{2/3} \left (-27 K[2]^2+3 \sqrt {3} \sqrt {K[2]^2 \left (27 K[2]^2-4\right )}+2\right )^{2/3}+4 \sqrt [3]{-27 K[2]^2+3 \sqrt {3} \sqrt {K[2]^2 \left (27 K[2]^2-4\right )}+2}+2 i \sqrt [3]{2} \sqrt {3}-2 \sqrt [3]{2}}dK[2]\&\right ]\left [\frac {x}{12}+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$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 ] \\ y(x)\to 0 \\ \end{align*}