34.2.4 problem 4

Internal problem ID [6034]
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 : Wednesday, March 05, 2025 at 12:10:09 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.006 (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: 1.374 (sec). Leaf size: 131

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 -\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: 0.859 (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)
 

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