89.7.9 problem 9

Internal problem ID [24440]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 4. Additional topics on equations of first order and first degree. Exercises at page 61
Problem number : 9
Date solved : Thursday, October 02, 2025 at 10:32:16 PM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} 6 y^{2}-x \left (2 x^{3}+y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.341 (sec). Leaf size: 64

ode:=6*y(x)^2-x*(2*x^3+y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x^{3} \left (x^{3}-\sqrt {x \left (x^{3}+8 c_1 \right )}\, x +4 c_1 \right )}{2 c_1} \\ y &= \frac {x^{3} \left (x^{3}+\sqrt {x \left (x^{3}+8 c_1 \right )}\, x +4 c_1 \right )}{2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 1.1 (sec). Leaf size: 123

ode=(6*y[x]^2)-x*( 2*x^3+y[x] )*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 2 x^3 \left (-1+\frac {2}{1-\frac {4 x^{3/2}}{\sqrt {16 x^3+c_1}}}\right )\\ y(x)&\to 2 x^3 \left (-1+\frac {2}{1+\frac {4 x^{3/2}}{\sqrt {16 x^3+c_1}}}\right )\\ y(x)&\to 0\\ y(x)&\to 2 x^3\\ y(x)&\to \frac {2 \left (\left (x^3\right )^{3/2}-x^{9/2}\right )}{x^{3/2}+\sqrt {x^3}} \end{align*}

✓ Sympy. Time used: 0.864 (sec). Leaf size: 78

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(2*x**3 + y(x))*Derivative(y(x), x) + 6*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \frac {3 C_{1} \sqrt {x^{9} \left (9 C_{1}^{2} x^{3} + 8\right )}}{2} + \frac {x^{3} \left (9 C_{1}^{2} x^{3} + 4\right )}{2}, \ y{\left (x \right )} = \frac {3 C_{1} \sqrt {x^{9} \left (9 C_{1}^{2} x^{3} + 8\right )}}{2} + \frac {x^{3} \left (9 C_{1}^{2} x^{3} + 4\right )}{2}\right ] \]