Internal problem ID [3667]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 32
Problem number: 941.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {y \left (y^{\prime }\right )^{2}-a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.11 (sec). Leaf size: 239
dsolve(y(x)*diff(y(x),x)^2 = a,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-12 c_{1} a^{2}+12 a^{2} x \right )^{\frac {2}{3}}}{4 a} \\ y \relax (x ) = \frac {\left (-\frac {\left (-12 c_{1} a^{2}+12 a^{2} x \right )^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, \left (-12 c_{1} a^{2}+12 a^{2} x \right )^{\frac {1}{3}}}{4}\right )^{2}}{a} \\ y \relax (x ) = \frac {\left (-\frac {\left (-12 c_{1} a^{2}+12 a^{2} x \right )^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, \left (-12 c_{1} a^{2}+12 a^{2} x \right )^{\frac {1}{3}}}{4}\right )^{2}}{a} \\ y \relax (x ) = \frac {\left (12 c_{1} a^{2}-12 a^{2} x \right )^{\frac {2}{3}}}{4 a} \\ y \relax (x ) = \frac {\left (-\frac {\left (12 c_{1} a^{2}-12 a^{2} x \right )^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, \left (12 c_{1} a^{2}-12 a^{2} x \right )^{\frac {1}{3}}}{4}\right )^{2}}{a} \\ y \relax (x ) = \frac {\left (-\frac {\left (12 c_{1} a^{2}-12 a^{2} x \right )^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, \left (12 c_{1} a^{2}-12 a^{2} x \right )^{\frac {1}{3}}}{4}\right )^{2}}{a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.149 (sec). Leaf size: 54
DSolve[y[x] (y'[x])^2==a,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (\frac {3}{2}\right )^{2/3} \left (-\sqrt {a} x+c_1\right ){}^{2/3} \\ y(x)\to \left (\frac {3}{2}\right )^{2/3} \left (\sqrt {a} x+c_1\right ){}^{2/3} \\ \end{align*}