58.7.6 problem 6

Internal problem ID [14661]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, Section 2.4. Special integrating factors and transformations. Exercises page 67
Problem number : 6
Date solved : Thursday, October 02, 2025 at 09:49:01 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} 8 x^{2} y^{3}-2 y^{4}+\left (5 x^{3} y^{2}-8 x y^{3}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.250 (sec). Leaf size: 34

ode:=8*x^2*y(x)^3-2*y(x)^4+(5*x^3*y(x)^2-8*x*y(x)^3)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= \operatorname {RootOf}\left (x^{6} \textit {\_Z}^{48}-x^{6} \textit {\_Z}^{30}-c_1 \right )^{18} x^{2} \\ \end{align*}

✓ Mathematica. Time used: 4.492 (sec). Leaf size: 411

ode=(8*x^2*y[x]^3-2*y[x]^4)+(5*x^3*y[x]^2-8*x*y[x]^3)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 0\\ y(x)&\to \text {Root}\left [-\text {$\#$1}^8+3 \text {$\#$1}^7 x^2-3 \text {$\#$1}^6 x^4+\text {$\#$1}^5 x^6+\frac {e^{18 c_1}}{x^2}\&,1\right ]\\ y(x)&\to \text {Root}\left [-\text {$\#$1}^8+3 \text {$\#$1}^7 x^2-3 \text {$\#$1}^6 x^4+\text {$\#$1}^5 x^6+\frac {e^{18 c_1}}{x^2}\&,2\right ]\\ y(x)&\to \text {Root}\left [-\text {$\#$1}^8+3 \text {$\#$1}^7 x^2-3 \text {$\#$1}^6 x^4+\text {$\#$1}^5 x^6+\frac {e^{18 c_1}}{x^2}\&,3\right ]\\ y(x)&\to \text {Root}\left [-\text {$\#$1}^8+3 \text {$\#$1}^7 x^2-3 \text {$\#$1}^6 x^4+\text {$\#$1}^5 x^6+\frac {e^{18 c_1}}{x^2}\&,4\right ]\\ y(x)&\to \text {Root}\left [-\text {$\#$1}^8+3 \text {$\#$1}^7 x^2-3 \text {$\#$1}^6 x^4+\text {$\#$1}^5 x^6+\frac {e^{18 c_1}}{x^2}\&,5\right ]\\ y(x)&\to \text {Root}\left [-\text {$\#$1}^8+3 \text {$\#$1}^7 x^2-3 \text {$\#$1}^6 x^4+\text {$\#$1}^5 x^6+\frac {e^{18 c_1}}{x^2}\&,6\right ]\\ y(x)&\to \text {Root}\left [-\text {$\#$1}^8+3 \text {$\#$1}^7 x^2-3 \text {$\#$1}^6 x^4+\text {$\#$1}^5 x^6+\frac {e^{18 c_1}}{x^2}\&,7\right ]\\ y(x)&\to \text {Root}\left [-\text {$\#$1}^8+3 \text {$\#$1}^7 x^2-3 \text {$\#$1}^6 x^4+\text {$\#$1}^5 x^6+\frac {e^{18 c_1}}{x^2}\&,8\right ]\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 6.928 (sec). Leaf size: 3

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(8*x**2*y(x)**3 + (5*x**3*y(x)**2 - 8*x*y(x)**3)*Derivative(y(x), x) - 2*y(x)**4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = 0 \]