28.2.4 problem 4

Internal problem ID [7169]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter 2, Equations of the first order and degree. page 20
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 04:24:48 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 54

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

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

✓ Mathematica. Time used: 4.003 (sec). Leaf size: 145

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

\begin{align*} y(x)&\to -\sqrt {-1+\exp \left (2 \left (\int _1^x\frac {1}{K[1]^3+K[1]}dK[1]+c_1\right )\right )}\\ y(x)&\to \sqrt {-1+\exp \left (2 \left (\int _1^x\frac {1}{K[1]^3+K[1]}dK[1]+c_1\right )\right )}\\ y(x)&\to -i\\ y(x)&\to i\\ y(x)&\to -\sqrt {\exp \left (2 \int _1^x\frac {1}{K[1]^3+K[1]}dK[1]\right )-1}\\ y(x)&\to \sqrt {\exp \left (2 \int _1^x\frac {1}{K[1]^3+K[1]}dK[1]\right )-1} \end{align*}

✓ Sympy. Time used: 0.548 (sec). Leaf size: 42

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

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