89.4.7 problem 8

Internal problem ID [24329]
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 39
Problem number : 8
Date solved : Thursday, October 02, 2025 at 10:18:20 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} \left (x^{3}-y^{5}\right ) y-x \left (x^{3}+y^{5}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 37

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

\[ \ln \left (x \right )-c_1 -\frac {5 \ln \left (\frac {y}{x^{{3}/{5}}}\right )}{2}+\frac {5 \ln \left (\frac {4 y^{5}-x^{3}}{x^{3}}\right )}{8} = 0 \]

✓ Mathematica. Time used: 1.582 (sec). Leaf size: 141

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

\begin{align*} y(x)&\to \text {Root}\left [4 \text {$\#$1}^5 x-4 \text {$\#$1}^4 c_1-x^4\&,1\right ]\\ y(x)&\to \text {Root}\left [4 \text {$\#$1}^5 x-4 \text {$\#$1}^4 c_1-x^4\&,2\right ]\\ y(x)&\to \text {Root}\left [4 \text {$\#$1}^5 x-4 \text {$\#$1}^4 c_1-x^4\&,3\right ]\\ y(x)&\to \text {Root}\left [4 \text {$\#$1}^5 x-4 \text {$\#$1}^4 c_1-x^4\&,4\right ]\\ y(x)&\to \text {Root}\left [4 \text {$\#$1}^5 x-4 \text {$\#$1}^4 c_1-x^4\&,5\right ] \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (x**3 - y(x)**5)*y(x)/(x*(x**3 + y(x)**5))