88.4.7 problem 7

Internal problem ID [23971]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 2. Differential equations of first order. Exercise at page 33
Problem number : 7
Date solved : Thursday, October 02, 2025 at 09:48:14 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 31

ode:=(x^2-1)*y(x)+(1+y(x)^2)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {\operatorname {LambertW}\left ({\mathrm e}^{-\frac {2}{3} x^{3}-2 c_1 +2 x}\right )}{2}-\frac {x^{3}}{3}-c_1 +x} \]

✓ Mathematica. Time used: 1.149 (sec). Leaf size: 60

ode=y[x]*(x^2-1)+(y[x]^2+1)*D[y[x],{x,1}]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\sqrt {W\left (e^{-\frac {2 x^3}{3}+2 x+2 c_1}\right )}\\ y(x)&\to \sqrt {W\left (e^{-\frac {2 x^3}{3}+2 x+2 c_1}\right )}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 3.370 (sec). Leaf size: 114

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

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