Internal problem ID [3719]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 33
Problem number: 995.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational]
Solve \begin {gather*} \boxed {9 y^{2} \left (y^{\prime }\right )^{2}-3 y^{\prime } x +y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.219 (sec). Leaf size: 122
dsolve(9*y(x)^2*diff(y(x),x)^2-3*x*diff(y(x),x)+y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{2} \\ y \relax (x ) = -\frac {2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{4}-\frac {i \sqrt {3}\, 2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{4} \\ y \relax (x ) = -\frac {2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{4}+\frac {i \sqrt {3}\, 2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{4} \\ y \relax (x ) = 0 \\ y \relax (x ) = \RootOf \left (-\ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {-6 \textit {\_a}^{3}+\frac {3 \sqrt {-4 \textit {\_a}^{3}+1}}{2}+\frac {3}{2}}{\textit {\_a} \left (4 \textit {\_a}^{3}-1\right )}d \textit {\_a} +c_{1}\right ) x^{\frac {2}{3}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.519 (sec). Leaf size: 244
DSolve[9 y[x]^2 (y'[x])^2 -3 x y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{\frac {c_1}{3}} \sqrt [3]{e^{c_1}-i x} \\ y(x)\to -\sqrt [3]{-1} e^{\frac {c_1}{3}} \sqrt [3]{e^{c_1}-i x} \\ y(x)\to (-1)^{2/3} e^{\frac {c_1}{3}} \sqrt [3]{e^{c_1}-i x} \\ y(x)\to e^{\frac {c_1}{3}} \sqrt [3]{i x+e^{c_1}} \\ y(x)\to -\sqrt [3]{-1} e^{\frac {c_1}{3}} \sqrt [3]{i x+e^{c_1}} \\ y(x)\to (-1)^{2/3} e^{\frac {c_1}{3}} \sqrt [3]{i x+e^{c_1}} \\ y(x)\to 0 \\ y(x)\to \left (-\frac {1}{2}\right )^{2/3} x^{2/3} \\ y(x)\to \frac {x^{2/3}}{2^{2/3}} \\ y(x)\to x^{2/3} \text {Root}\left [4 \text {$\#$1}^3-1\&,2\right ] \\ \end{align*}