29.33.32 problem 995

Internal problem ID [5571]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 33
Problem number : 995
Date solved : Sunday, March 30, 2025 at 09:01:53 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational]

\begin{align*} 9 y^{2} {y^{\prime }}^{2}-3 x y^{\prime }+y&=0 \end{align*}

✓ Maple. Time used: 0.105 (sec). Leaf size: 112

ode:=9*y(x)^2*diff(y(x),x)^2-3*x*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {2^{{1}/{3}} \left (x^{2}\right )^{{1}/{3}}}{2} \\ y &= -\frac {2^{{1}/{3}} \left (x^{2}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{4} \\ y &= \frac {2^{{1}/{3}} \left (x^{2}\right )^{{1}/{3}} \left (-1+i \sqrt {3}\right )}{4} \\ y &= 0 \\ y &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+3 \int _{}^{\textit {\_Z}}-\frac {4 \textit {\_a}^{3}-\sqrt {-4 \textit {\_a}^{3}+1}-1}{\textit {\_a} \left (4 \textit {\_a}^{3}-1\right )}d \textit {\_a} +2 c_1 \right ) x^{{2}/{3}} \\ \end{align*}

✓ Mathematica. Time used: 0.498 (sec). Leaf size: 238

ode=9 y[x]^2 (D[y[x],x])^2 -3 x D[y[x],x]+y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\sqrt [3]{-2} \sqrt [3]{e^{c_1} \left (x-2 e^{c_1}\right )} \\ y(x)\to \sqrt [3]{2} \sqrt [3]{e^{c_1} \left (x-2 e^{c_1}\right )} \\ y(x)\to (-1)^{2/3} \sqrt [3]{2} \sqrt [3]{e^{c_1} \left (x-2 e^{c_1}\right )} \\ y(x)\to \frac {\sqrt [3]{-e^{c_1} \left (-2 x+e^{c_1}\right )}}{2^{2/3}} \\ y(x)\to -\frac {\sqrt [3]{-1} \sqrt [3]{-e^{c_1} \left (-2 x+e^{c_1}\right )}}{2^{2/3}} \\ y(x)\to \left (-\frac {1}{2}\right )^{2/3} \sqrt [3]{-e^{c_1} \left (-2 x+e^{c_1}\right )} \\ 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 -\frac {\sqrt [3]{-1} x^{2/3}}{2^{2/3}} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*x*Derivative(y(x), x) + 9*y(x)**2*Derivative(y(x), x)**2 + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(x + sqrt(x**2 - 4*y(x)**3))/(6*y(x)**2) + Derivative(y(x), x) cannot be solved by the factorable group method