30.4.1 problem 1

Internal problem ID [7484]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.5, Special Integrating Factors. Exercises. page 69
Problem number : 1
Date solved : Tuesday, September 30, 2025 at 04:39:18 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class B`]]

\begin{align*} 2 x +\frac {y}{x}+\left (x y-1\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 51

ode:=2*x+y(x)/x+(x*y(x)-1)*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: 0.515 (sec). Leaf size: 68

ode=( 2*x+y[x]/x )+( x*y[x]-1  )*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: 2.102 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x + (x*y(x) - 1)*Derivative(y(x), x) + y(x)/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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