89.2.10 problem 10

Internal problem ID [24275]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 27
Problem number : 10
Date solved : Thursday, October 02, 2025 at 10:05:39 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} x \left (x^{2}+y^{2}\right )^{2} \left (y-x y^{\prime }\right )+y^{6} y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 44

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

\[ y = {\mathrm e}^{\operatorname {RootOf}\left (6 \,{\mathrm e}^{6 \textit {\_Z}} \ln \left (x \right )+6 \,{\mathrm e}^{6 \textit {\_Z}} c_1 +6 \textit {\_Z} \,{\mathrm e}^{6 \textit {\_Z}}+3 \,{\mathrm e}^{4 \textit {\_Z}}+3 \,{\mathrm e}^{2 \textit {\_Z}}+1\right )} x \]

✓ Mathematica. Time used: 0.125 (sec). Leaf size: 52

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

\[ \text {Solve}\left [\frac {x^6}{6 y(x)^6}+\frac {x^4}{2 y(x)^4}+\frac {x^2}{2 y(x)^2}+\log \left (\frac {y(x)}{x}\right )=-\log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.321 (sec). Leaf size: 36

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

\[ \log {\left (y{\left (x \right )} \right )} = C_{1} - \frac {x^{6}}{6 y^{6}{\left (x \right )}} - \frac {x^{4}}{2 y^{4}{\left (x \right )}} - \frac {x^{2}}{2 y^{2}{\left (x \right )}} \]