Internal problem ID [4228]
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]
\[ \boxed {\left (2-3 y\right )^{2} {y^{\prime }}^{2}+4 y=4} \]
✓ Solution by Maple
Time used: 0.078 (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 \left (x \right ) = 1 y \left (x \right ) = -{\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 \left (x \right ) = -{\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 \left (x \right ) = -{\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 \left (x \right ) = -{\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 \left (x \right ) = -{\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 \left (x \right ) = -{\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: 4.408 (sec). Leaf size: 896
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}{12} \left (2 \sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (108 x^2+108 c_1 x-16+27 c_1{}^2\right )}-108 c_1 x+8-27 c_1{}^2}+\frac {8}{\sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (108 x^2+108 c_1 x-16+27 c_1{}^2\right )}-108 c_1 x+8-27 c_1{}^2}}+4\right ) y(x)\to \frac {1}{24} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (108 x^2+108 c_1 x-16+27 c_1{}^2\right )}-108 c_1 x+8-27 c_1{}^2}-\frac {8 \left (1+i \sqrt {3}\right )}{\sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (108 x^2+108 c_1 x-16+27 c_1{}^2\right )}-108 c_1 x+8-27 c_1{}^2}}+8\right ) y(x)\to \frac {1}{24} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (108 x^2+108 c_1 x-16+27 c_1{}^2\right )}-108 c_1 x+8-27 c_1{}^2}+\frac {8 i \left (\sqrt {3}+i\right )}{\sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (108 x^2+108 c_1 x-16+27 c_1{}^2\right )}-108 c_1 x+8-27 c_1{}^2}}+8\right ) y(x)\to \frac {1}{12} \left (2 \sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (108 x^2-108 c_1 x-16+27 c_1{}^2\right )}+108 c_1 x+8-27 c_1{}^2}+\frac {8}{\sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (108 x^2-108 c_1 x-16+27 c_1{}^2\right )}+108 c_1 x+8-27 c_1{}^2}}+4\right ) y(x)\to \frac {1}{24} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (108 x^2-108 c_1 x-16+27 c_1{}^2\right )}+108 c_1 x+8-27 c_1{}^2}+\frac {-8-8 i \sqrt {3}}{\sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (108 x^2-108 c_1 x-16+27 c_1{}^2\right )}+108 c_1 x+8-27 c_1{}^2}}+8\right ) y(x)\to \frac {1}{24} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (108 x^2-108 c_1 x-16+27 c_1{}^2\right )}+108 c_1 x+8-27 c_1{}^2}+\frac {-8+8 i \sqrt {3}}{\sqrt [3]{-108 x^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (108 x^2-108 c_1 x-16+27 c_1{}^2\right )}+108 c_1 x+8-27 c_1{}^2}}+8\right ) y(x)\to 1 \end{align*}