77.5.15 problem 15

Internal problem ID [20395]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Exercise II (D) at page 16
Problem number : 15
Date solved : Thursday, October 02, 2025 at 05:50:52 PM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 27

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

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

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 33

ode=D[y[x],x]+x/(1+x^2)*y[x]==1/(2*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 )-2 c_1}{2 \sqrt {x^2+1}} \end{align*}

✓ Sympy. Time used: 1.803 (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/(2*x*(x**2 + 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {C_{1} + \frac {\log {\left (\sqrt {x^{2} + 1} - 1 \right )}}{4} - \frac {\log {\left (\sqrt {x^{2} + 1} + 1 \right )}}{4}}{\sqrt {x^{2} + 1}} \]