23.1.92 problem 88 (a)

Internal problem ID [4699]
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 : 88 (a)
Date solved : Tuesday, September 30, 2025 at 08:15:28 AM
CAS classification : [_Bernoulli]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 53

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

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

✓ Mathematica. Time used: 7.428 (sec). Leaf size: 70

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

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

✓ Sympy. Time used: 0.713 (sec). Leaf size: 63

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

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