75.3.5 problem 5

Internal problem ID [19909]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter III. Ordinary differential equations of the first order and first degree. Exercises at page 33
Problem number : 5
Date solved : Thursday, October 02, 2025 at 05:00:50 PM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 24

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

\[ y = \frac {-\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right )+c_1}{\sqrt {x^{2}+1}} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 30

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

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

✓ Sympy. Time used: 1.782 (sec). Leaf size: 39

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

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