89.32.13 problem 15

Internal problem ID [24973]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 16. Equations of order one and higher degree. Miscellaneous Exercises at page 246
Problem number : 15
Date solved : Thursday, October 02, 2025 at 11:45:13 PM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} 16 {y^{\prime }}^{2} x +8 y y^{\prime }+y^{6}&=0 \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 103

ode:=16*x*diff(y(x),x)^2+8*y(x)*diff(y(x),x)+y(x)^6 = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✗ Mathematica

ode=16*x*D[y[x],x]^2+8*x*y[x]*D[y[x],x]+y[x]^6==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(8*x*y(x)*Derivative(y(x), x) + 16*x*Derivative(y(x), x)**2 + y(x)**6,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-x + sqrt(x*(x - y(x)**4)))*y(x)/(4*x) ca