81.5.8 problem 6-8

Internal problem ID [21546]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 6. Method of grouping. Page 96.
Problem number : 6-8
Date solved : Thursday, October 02, 2025 at 07:47:39 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 51

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

\begin{align*} y &= \frac {c_1 x -\sqrt {-3 x^{2} c_1^{2}+4}}{2 c_1} \\ y &= \frac {c_1 x +\sqrt {-3 x^{2} c_1^{2}+4}}{2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.286 (sec). Leaf size: 106

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

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

✓ Sympy. Time used: 0.815 (sec). Leaf size: 34

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

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