89.32.3 problem 3

Internal problem ID [24963]
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 : 3
Date solved : Thursday, October 02, 2025 at 11:45:04 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} 9 {y^{\prime }}^{2}+3 x y^{4} y^{\prime }+y^{5}&=0 \end{align*}

✓ Maple. Time used: 0.171 (sec). Leaf size: 102

ode:=9*diff(y(x),x)^2+3*x*y(x)^4*diff(y(x),x)+y(x)^5 = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.41 (sec). Leaf size: 216

ode=9*D[y[x],x]^2+3*x*y[x]^4*D[y[x],x]+y[x]^5==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [-\frac {\sqrt {4-x^2 y(x)^3} y(x)^4 \text {arcsinh}\left (\frac {1}{2} x \sqrt {-y(x)^3}\right )}{\sqrt {-y(x)^3} \sqrt {y(x)^5 \left (x^2 y(x)^3-4\right )}}-\frac {3}{2} \log (y(x))=c_1,y(x)\right ]\\ \text {Solve}\left [\frac {y(x)^4 \sqrt {4-x^2 y(x)^3} \text {arcsinh}\left (\frac {1}{2} x \sqrt {-y(x)^3}\right )}{\sqrt {-y(x)^3} \sqrt {y(x)^5 \left (x^2 y(x)^3-4\right )}}-\frac {3}{2} \log (y(x))=c_1,y(x)\right ]\\ y(x)&\to 0\\ y(x)&\to \frac {(-2)^{2/3}}{x^{2/3}}\\ y(x)&\to \frac {2^{2/3}}{x^{2/3}}\\ y(x)&\to -\frac {\sqrt [3]{-1} 2^{2/3}}{x^{2/3}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x*y(x)**4*Derivative(y(x), x) + y(x)**5 + 9*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE x*y(x)**4/6 - sqrt((x**2*y(x)**3 - 4)*y(x)**5)/6 + Derivative(y(