47.4.38 problem 41

Internal problem ID [9812]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing. EXERCISES Page 324
Problem number : 41
Date solved : Tuesday, September 30, 2025 at 06:42:26 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} {y^{\prime \prime }}^{3}&=12 y^{\prime } \left (x y^{\prime \prime }-2 y^{\prime }\right ) \end{align*}

✓ Maple. Time used: 3.454 (sec). Leaf size: 174

ode:=diff(diff(y(x),x),x)^3 = 12*diff(y(x),x)*(x*diff(diff(y(x),x),x)-2*diff(y(x),x)); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x^{4}}{9}+c_1 \\ y &= c_1 \\ y &= \int \operatorname {RootOf}\left (-6 \ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {3 \textit {\_f} \sqrt {\frac {1}{\textit {\_f} \left (9 \textit {\_f} -4\right )}}\, 2^{{1}/{3}} \left (\left (3 \sqrt {\frac {1}{\textit {\_f} \left (9 \textit {\_f} -4\right )}}\, \textit {\_f} +1\right )^{2} \left (9 \textit {\_f} -4\right )^{4}\right )^{{1}/{3}}-2 \,2^{{2}/{3}} \left (\left (3 \sqrt {\frac {1}{\textit {\_f} \left (9 \textit {\_f} -4\right )}}\, \textit {\_f} +1\right ) \left (9 \textit {\_f} -4\right )^{2}\right )^{{1}/{3}}-2^{{1}/{3}} \left (\left (3 \sqrt {\frac {1}{\textit {\_f} \left (9 \textit {\_f} -4\right )}}\, \textit {\_f} +1\right )^{2} \left (9 \textit {\_f} -4\right )^{4}\right )^{{1}/{3}}+18 \textit {\_f} -8}{\textit {\_f} \left (9 \textit {\_f} -4\right )}d \textit {\_f} +6 c_1 \right ) x^{3}d x +c_2 \\ \end{align*}

✗ Mathematica

ode=(D[y[x],{x,2}])^3==12*D[y[x],x]*(x*D[y[x],{x,2}]-2*D[y[x],x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-12*x*Derivative(y(x), (x, 2)) + 24*Derivative(y(x), x))*Derivative(y(x), x) + Derivative(y(x), (x, 2))**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out