23.1.314 problem 302

Internal problem ID [4921]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 302
Date solved : Tuesday, September 30, 2025 at 08:58:53 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 17

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

\[ y = \frac {1}{1+\sqrt {x^{2}+1}\, c_1} \]

✓ Mathematica. Time used: 0.208 (sec). Leaf size: 49

ode=(1+x^2)*D[y[x],x]+x*y[x]*(1-y[x])==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]-1) K[1]}dK[1]\&\right ]\left [\frac {1}{2} \log \left (x^2+1\right )+c_1\right ]\\ y(x)&\to 0\\ y(x)&\to 1 \end{align*}

✓ Sympy. Time used: 0.868 (sec). Leaf size: 46

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

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