81.1.22 problem 2-20

Internal problem ID [21467]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 2. Separable differential equations
Problem number : 2-20
Date solved : Thursday, October 02, 2025 at 07:39:04 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {x}{y^{2} \sqrt {x^{2}+1}} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 69

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

\begin{align*} y &= \left (3 \sqrt {x^{2}+1}+c_1 \right )^{{1}/{3}} \\ y &= -\frac {\left (3 \sqrt {x^{2}+1}+c_1 \right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2} \\ y &= \frac {\left (3 \sqrt {x^{2}+1}+c_1 \right )^{{1}/{3}} \left (-1+i \sqrt {3}\right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 1.793 (sec). Leaf size: 85

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

\begin{align*} y(x)&\to -\sqrt [3]{-3} \sqrt [3]{\sqrt {x^2+1}+c_1}\\ y(x)&\to \sqrt [3]{3} \sqrt [3]{\sqrt {x^2+1}+c_1}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{3} \sqrt [3]{\sqrt {x^2+1}+c_1} \end{align*}

✗ Sympy

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

Timed Out