69.20.30 problem 669

Internal problem ID [18451]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.5 Linear equations with variable coefficients. The Lagrange method. Exercises page 148
Problem number : 669
Date solved : Thursday, October 02, 2025 at 03:12:10 PM
CAS classification : [[_2nd_order, _missing_y]]

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

With initial conditions

\begin{align*} y \left (\infty \right )&=\frac {\pi ^{2}}{8} \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.082 (sec). Leaf size: 10

ode:=(x^2+1)*diff(diff(y(x),x),x)+2*x*diff(y(x),x) = 1/(x^2+1); 
ic:=[y(infinity) = 1/8*Pi^2, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\arctan \left (x \right )^{2}}{2} \]

✗ Mathematica

ode=(1+x^2)*D[y[x],{x,2}]+2*x*D[y[x],x]==1/(1+x^2); 
ic={y[Infinity]==Pi^2/8,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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✓ Sympy. Time used: 0.581 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), x) + (x**2 + 1)*Derivative(y(x), (x, 2)) - 1/(x**2 + 1),0) 
ics = {y(oo): pi**2/8, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} - \frac {\log {\left (x - i \right )}^{2}}{8} + \frac {\log {\left (x - i \right )} \log {\left (x + i \right )}}{4} - \frac {\log {\left (x + i \right )}^{2}}{8} + \frac {\pi \operatorname {atan}{\left (x \right )}}{2} \]