23.1.190 problem 190

Internal problem ID [4797]
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 : 190
Date solved : Tuesday, September 30, 2025 at 08:40:48 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 30

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

\begin{align*} y &= \frac {x}{\sqrt {-x^{2}+c_1}} \\ y &= -\frac {x}{\sqrt {-x^{2}+c_1}} \\ \end{align*}

✓ Mathematica. Time used: 0.369 (sec). Leaf size: 110

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

\begin{align*} y(x)&\to -\frac {i e^{c_1} x}{\sqrt {-1+e^{2 c_1} x^2}}\\ y(x)&\to \frac {i e^{c_1} x}{\sqrt {-1+e^{2 c_1} x^2}}\\ y(x)&\to 0\\ y(x)&\to -i\\ y(x)&\to i\\ y(x)&\to -\frac {i x}{\sqrt {x^2}}\\ y(x)&\to \frac {i x}{\sqrt {x^2}} \end{align*}

✓ Sympy. Time used: 0.533 (sec). Leaf size: 36

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

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