75.20.30 problem 669

Internal problem ID [17164]
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 : Tuesday, January 28, 2025 at 09:55:45 AM
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*}

✓ Solution by Maple

Time used: 0.203 (sec). Leaf size: 10

dsolve([(1+x^2)*diff(y(x),x$2)+2*x*diff(y(x),x)=1/(1+x^2),y(infinity) = 1/8*Pi^2, D(y)(0) = 0],y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.048 (sec). Leaf size: 13

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

\[ y(x)\to \frac {\arctan (x)^2}{2} \]