71.1.19 problem 35 (page 40)

Internal problem ID [19195]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 35 (page 40)
Date solved : Thursday, October 02, 2025 at 03:43:00 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: 0.001 (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: 0.398 (sec). Leaf size: 56

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

\begin{align*} y(x)&\to e^{-\frac {1}{4} x^2 \left (x^2+2\right )} \left (\int _1^x\frac {e^{\frac {1}{4} K[1]^2 \left (K[1]^2+2\right )}}{K[1]^3+K[1]}dK[1]+c_1\right ) \end{align*}

✓ Sympy. Time used: 1.784 (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)
 

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