Internal problem ID [3720]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 34
Problem number: 996.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {\left (2-3 y\right )^{2} \left (y^{\prime }\right )^{2}-4+4 y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.132 (sec). Leaf size: 713
dsolve((2-3*y(x))^2*diff(y(x),x)^2 = 4-4*y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = 1 \\ y \relax (x ) = -\left (\frac {\left (108 c_{1}-108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108 c_{1}-108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}\right )^{2}+1 \\ y \relax (x ) = -\left (-\frac {\left (108 c_{1}-108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}{12}-\frac {1}{\left (108 c_{1}-108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (108 c_{1}-108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108 c_{1}-108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+1 \\ y \relax (x ) = -\left (-\frac {\left (108 c_{1}-108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}{12}-\frac {1}{\left (108 c_{1}-108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (108 c_{1}-108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108 c_{1}-108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+1 \\ y \relax (x ) = -\left (\frac {\left (-108 c_{1}+108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (-108 c_{1}+108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}\right )^{2}+1 \\ y \relax (x ) = -\left (-\frac {\left (-108 c_{1}+108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}{12}-\frac {1}{\left (-108 c_{1}+108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-108 c_{1}+108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (-108 c_{1}+108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+1 \\ y \relax (x ) = -\left (-\frac {\left (-108 c_{1}+108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}{12}-\frac {1}{\left (-108 c_{1}+108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-108 c_{1}+108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (-108 c_{1}+108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+1 \\ \end{align*}
✓ Solution by Mathematica
Time used: 5.079 (sec). Leaf size: 746
DSolve[(2-3 y[x])^2 (y'[x])^2 ==4(1-y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{6} \left (\sqrt [3]{-27 (2 x+c_1){}^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (-16+27 (2 x+c_1){}^2\right )}+8}+\frac {4}{\sqrt [3]{-27 (2 x+c_1){}^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (-16+27 (2 x+c_1){}^2\right )}+8}}+2\right ) \\ y(x)\to \frac {1}{24} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{-27 (2 x+c_1){}^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (-16+27 (2 x+c_1){}^2\right )}+8}+\frac {-8-8 i \sqrt {3}}{\sqrt [3]{-27 (2 x+c_1){}^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (-16+27 (2 x+c_1){}^2\right )}+8}}+8\right ) \\ y(x)\to \frac {1}{24} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{-27 (2 x+c_1){}^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (-16+27 (2 x+c_1){}^2\right )}+8}+\frac {-8+8 i \sqrt {3}}{\sqrt [3]{-27 (2 x+c_1){}^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (-16+27 (2 x+c_1){}^2\right )}+8}}+8\right ) \\ y(x)\to \frac {1}{6} \left (\sqrt [3]{-27 (-2 x+c_1){}^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (-16+27 (-2 x+c_1){}^2\right )}+8}+\frac {4}{\sqrt [3]{-27 (-2 x+c_1){}^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (-16+27 (-2 x+c_1){}^2\right )}+8}}+2\right ) \\ y(x)\to \frac {1}{24} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{-27 (-2 x+c_1){}^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (-16+27 (-2 x+c_1){}^2\right )}+8}+\frac {-8-8 i \sqrt {3}}{\sqrt [3]{-27 (-2 x+c_1){}^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (-16+27 (-2 x+c_1){}^2\right )}+8}}+8\right ) \\ y(x)\to \frac {1}{24} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{-27 (-2 x+c_1){}^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (-16+27 (-2 x+c_1){}^2\right )}+8}+\frac {-8+8 i \sqrt {3}}{\sqrt [3]{-27 (-2 x+c_1){}^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (-16+27 (-2 x+c_1){}^2\right )}+8}}+8\right ) \\ y(x)\to 1 \\ \end{align*}