89.9.12 problem 12

Internal problem ID [24492]
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 72
Problem number : 12
Date solved : Thursday, October 02, 2025 at 10:42:35 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y \left (8 x -9 y\right )+2 x \left (x -3 y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 69

ode:=y(x)*(8*x-9*y(x))+2*x*(x-3*y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x^{3} c_1^{3}-\sqrt {c_1 x \left (c_1^{5} x^{5}+3\right )}}{3 x^{2} c_1^{3}} \\ y &= \frac {x^{3} c_1^{3}+\sqrt {c_1 x \left (c_1^{5} x^{5}+3\right )}}{3 x^{2} c_1^{3}} \\ \end{align*}

✓ Mathematica. Time used: 0.466 (sec). Leaf size: 118

ode=y[x]*( 8*x-9*y[x] )+2*x*( x-3*y[x]   )*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{3} \left (x-\frac {\sqrt {x^5-3 e^{2 c_1}}}{x^{3/2}}\right )\\ y(x)&\to \frac {1}{3} \left (x+\frac {\sqrt {x^5-3 e^{2 c_1}}}{x^{3/2}}\right )\\ y(x)&\to \frac {x}{3}-\frac {\sqrt {x^5}}{3 x^{3/2}}\\ y(x)&\to \frac {x^{5/2}+\sqrt {x^5}}{3 x^{3/2}} \end{align*}

✓ Sympy. Time used: 1.029 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*(x - 3*y(x))*Derivative(y(x), x) + (8*x - 9*y(x))*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {x \left (1 - \sqrt {\frac {C_{1}}{x^{5}} + 1}\right )}{3}, \ y{\left (x \right )} = \frac {x \left (\sqrt {\frac {C_{1}}{x^{5}} + 1} + 1\right )}{3}\right ] \]