41.1.7 problem 7

Internal problem ID [8674]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.1 Separable equations problems. page 7
Problem number : 7
Date solved : Tuesday, September 30, 2025 at 05:40:43 PM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.101 (sec). Leaf size: 19

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

\[ y = \frac {1}{-i \pi +\ln \left (x -1\right )+\ln \left (x +1\right )+1} \]

✓ Mathematica. Time used: 0.112 (sec). Leaf size: 26

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

\begin{align*} y(x)&\to \frac {i}{i \log \left (x^2-1\right )+\pi +i} \end{align*}

✓ Sympy. Time used: 0.122 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = - \frac {1}{- \log {\left (x^{2} - 1 \right )} - 1 + i \pi } \]