86.1.12 problem 12

Internal problem ID [23074]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 3. Some standard types of differential equations. Exercise 3b at page 43
Problem number : 12
Date solved : Thursday, October 02, 2025 at 09:18:56 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 54

ode:=x*y(x)*(x^2+1)*diff(y(x),x)-y(x)^2 = 1; 
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: 0.728 (sec). Leaf size: 131

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

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

✓ Sympy. Time used: 0.575 (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)
 

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