23.1.90 problem 84

Internal problem ID [4697]
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 : 84
Date solved : Tuesday, September 30, 2025 at 08:15:23 AM
CAS classification : [_Bernoulli]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 39

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

\begin{align*} y &= -\frac {2}{\sqrt {2+4 \,{\mathrm e}^{2 x} c_1 +4 x}} \\ y &= \frac {2}{\sqrt {2+4 \,{\mathrm e}^{2 x} c_1 +4 x}} \\ \end{align*}

✓ Mathematica. Time used: 9.481 (sec). Leaf size: 78

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

\begin{align*} y(x)&\to -\frac {1}{\sqrt {e^{2 x} \left (-2 \int _1^xe^{-2 K[1]} K[1]dK[1]+c_1\right )}}\\ y(x)&\to \frac {1}{\sqrt {e^{2 x} \left (-2 \int _1^xe^{-2 K[1]} K[1]dK[1]+c_1\right )}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.724 (sec). Leaf size: 49

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

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