81.4.20 problem 5-21

Internal problem ID [21534]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 5. Integrating factors. Page 72.
Problem number : 5-21
Date solved : Thursday, October 02, 2025 at 07:47:00 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 51

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

\begin{align*} y &= \frac {1+\sqrt {-2 c_1 \,x^{2}-4 x^{3}+1}}{x} \\ y &= \frac {1-\sqrt {-2 c_1 \,x^{2}-4 x^{3}+1}}{x} \\ \end{align*}

✓ Mathematica. Time used: 0.39 (sec). Leaf size: 68

ode=(2*x^2+y[x])+(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}{x}-\sqrt {\frac {1}{x^2}} \sqrt {-4 x^3+c_1 x^2+1}\\ y(x)&\to \frac {1}{x}+\sqrt {\frac {1}{x^2}} \sqrt {-4 x^3+c_1 x^2+1} \end{align*}

✓ Sympy. Time used: 2.123 (sec). Leaf size: 41

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

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