6.5.8 problem 4

Internal problem ID [1632]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable Equations. Section 2.4 Page 68
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 04:40:54 AM
CAS classification : [_rational, _Bernoulli]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 38

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

\begin{align*} y &= \frac {\sqrt {2 x +c_1}}{x^{2}+1} \\ y &= -\frac {\sqrt {2 x +c_1}}{x^{2}+1} \\ \end{align*}

✓ Mathematica. Time used: 0.169 (sec). Leaf size: 46

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

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

✓ Sympy. Time used: 0.338 (sec). Leaf size: 32

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

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