6.3.15 problem 16

Internal problem ID [1592]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. separable equations. Section 2.2 Page 52
Problem number : 16
Date solved : Tuesday, September 30, 2025 at 04:38:50 AM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.113 (sec). Leaf size: 16

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

\[ y = \frac {1}{\sqrt {2 \,{\mathrm e}^{-2 x^{2}}-1}} \]

✓ Mathematica. Time used: 60.063 (sec). Leaf size: 27

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

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

✗ Sympy

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

NotImplementedError : Initial conditions produced too many solutions for constants