26.6.34 problem Exercise 12.34, page 103

Internal problem ID [7034]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 12, Miscellaneous Methods
Problem number : Exercise 12.34, page 103
Date solved : Tuesday, September 30, 2025 at 04:16:40 PM
CAS classification : [_separable]

\begin{align*} \left (x^{2}-1\right ) y^{\prime }+x y-3 x y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 20

ode:=(x^2-1)*diff(y(x),x)+x*y(x)-3*x*y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {1}{3+\sqrt {x -1}\, \sqrt {x +1}\, c_1} \]

✓ Mathematica. Time used: 0.234 (sec). Leaf size: 53

ode=(x^2-1)*D[y[x],x]+x*y[x]-3*x*y[x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] (3 K[1]-1)}dK[1]\&\right ]\left [\frac {1}{2} \log \left (x^2-1\right )+c_1\right ]\\ y(x)&\to 0\\ y(x)&\to \frac {1}{3} \end{align*}

✓ Sympy. Time used: 1.038 (sec). Leaf size: 53

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*x*y(x)**2 + x*y(x) + (x**2 - 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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