Internal problem ID [3760]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 35
Problem number: 1047.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{3}-y \left (y^{\prime }\right )^{2}+y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.297 (sec). Leaf size: 423
dsolve(diff(y(x),x)^3-y(x)*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 (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{4 \textit {\_a}^{2}+2 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-3 \textit {\_a}^{4} \left (-27+4 \textit {\_a} \right )}\right )^{\frac {1}{3}}+\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}}d \textit {\_a} \right )-c_{1} = 0 \\ x -\left (\int _{}^{y \relax (x )}-\frac {12 \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, \textit {\_a}^{2}-4 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-3 \textit {\_a}^{4} \left (-27+4 \textit {\_a} \right )}\right )^{\frac {1}{3}}+\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}+4 \textit {\_a}^{2}}d \textit {\_a} \right )-c_{1} = 0 \\ x -\left (\int _{}^{y \relax (x )}\frac {12 \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, \textit {\_a}^{2}+4 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-3 \textit {\_a}^{4} \left (-27+4 \textit {\_a} \right )}\right )^{\frac {1}{3}}-\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}-4 \textit {\_a}^{2}}d \textit {\_a} \right )-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 41.755 (sec). Leaf size: 653
DSolve[(y'[x])^3 -y[x] (y'[x])^2+y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{2 K[1]^3-27 K[1]^2+3 \sqrt {3} \sqrt {-K[1]^4 (4 K[1]-27)}}}{2 \sqrt [3]{2} K[1]^2+2 \sqrt [3]{2 K[1]^3-27 K[1]^2+3 \sqrt {3} \sqrt {-K[1]^4 (4 K[1]-27)}} K[1]+2^{2/3} \left (2 K[1]^3-27 K[1]^2+3 \sqrt {3} \sqrt {-K[1]^4 (4 K[1]-27)}\right )^{2/3}}dK[1]\&\right ]\left [\frac {x}{6}+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{2 K[2]^3-27 K[2]^2+3 \sqrt {3} \sqrt {-K[2]^4 (4 K[2]-27)}}}{2 i \sqrt [3]{2} \sqrt {3} K[2]^2-2 \sqrt [3]{2} K[2]^2+4 \sqrt [3]{2 K[2]^3-27 K[2]^2+3 \sqrt {3} \sqrt {-K[2]^4 (4 K[2]-27)}} K[2]-i 2^{2/3} \sqrt {3} \left (2 K[2]^3-27 K[2]^2+3 \sqrt {3} \sqrt {-K[2]^4 (4 K[2]-27)}\right )^{2/3}-2^{2/3} \left (2 K[2]^3-27 K[2]^2+3 \sqrt {3} \sqrt {-K[2]^4 (4 K[2]-27)}\right )^{2/3}}dK[2]\&\right ]\left [\frac {x}{12}+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{2 K[3]^3-27 K[3]^2+3 \sqrt {3} \sqrt {-K[3]^4 (4 K[3]-27)}}}{-2 i \sqrt [3]{2} \sqrt {3} K[3]^2-2 \sqrt [3]{2} K[3]^2+4 \sqrt [3]{2 K[3]^3-27 K[3]^2+3 \sqrt {3} \sqrt {-K[3]^4 (4 K[3]-27)}} K[3]+i 2^{2/3} \sqrt {3} \left (2 K[3]^3-27 K[3]^2+3 \sqrt {3} \sqrt {-K[3]^4 (4 K[3]-27)}\right )^{2/3}-2^{2/3} \left (2 K[3]^3-27 K[3]^2+3 \sqrt {3} \sqrt {-K[3]^4 (4 K[3]-27)}\right )^{2/3}}dK[3]\&\right ]\left [\frac {x}{12}+c_1\right ] \\ y(x)\to 0 \\ \end{align*}