23.1.247 problem 242

Internal problem ID [4854]
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 : 242
Date solved : Tuesday, September 30, 2025 at 08:45:05 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 23

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

\begin{align*} y &= \frac {1}{\sqrt {c_1 x -1}} \\ y &= -\frac {1}{\sqrt {c_1 x -1}} \\ \end{align*}

✓ Mathematica. Time used: 0.227 (sec). Leaf size: 72

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

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

✓ Sympy. Time used: 0.496 (sec). Leaf size: 22

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

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