Internal problem ID [4263]
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]
\[ \boxed {{y^{\prime }}^{3}-{y^{\prime }}^{2}+y^{2}=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 465
dsolve(diff(y(x),x)^3-diff(y(x),x)^2+y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ -3 \,3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (\int _{}^{y \left (x \right )}\frac {\left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}+3^{\frac {2}{3}} 2^{\frac {1}{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {2}{3}}+6}d \textit {\_a} \right )+x -c_{1} &= 0 \\ \frac {36 \,3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (\int _{}^{y \left (x \right )}\frac {\left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{\left (3 i \sqrt {3}+3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}+3\right ) \left (3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}-6\right )}d \textit {\_a} \right )+\left (x -c_{1} \right ) \left (1+i \sqrt {3}\right )}{1+i \sqrt {3}} &= 0 \\ \frac {i \left (x -c_{1} \right ) \sqrt {3}+36 \,3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (\int _{}^{y \left (x \right )}\frac {\left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{\left (-3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}+6\right ) \left (-3 i \sqrt {3}+3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}+3\right )}d \textit {\_a} \right )-x +c_{1}}{-1+i \sqrt {3}} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 47.889 (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*}