Internal problem ID [5342]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 10. Singular solutions, Extraneous loci. Supplemetary problems. Page
74
Problem number: 14.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
\[ \boxed {\left (3 y-1\right )^{2} {y^{\prime }}^{2}-4 y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 689
dsolve((3*y(x)-1)^2*diff(y(x),x)^2=4*y(x),y(x), singsol=all)
\begin{align*} y = 0 y = {\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} y = {\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} y = {\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} y = {\left (\frac {\left (108 x -108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108 x -108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}\right )}^{2} y = {\left (-\frac {\left (108 x -108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}{12}-\frac {1}{\left (108 x -108 c_{1} +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 x -108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108 x -108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}\right )}{2}\right )}^{2} y = {\left (-\frac {\left (108 x -108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}{12}-\frac {1}{\left (108 x -108 c_{1} +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 x -108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108 x -108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}\right )}{2}\right )}^{2} \end{align*}
✓ Solution by Mathematica
Time used: 4.472 (sec). Leaf size: 892
DSolve[(3*y[x]-1)^2*y'[x]^2==4*y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\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}\right ){}^2}{6 \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}} 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}}+16\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}}+16\right ) y(x)\to \frac {\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}\right ){}^2}{6 \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}} 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}}+16\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}}+16\right ) y(x)\to 0 \end{align*}