Internal problem ID [15434]
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 ODE’s). Section 15.5 Linear equations with variable coefficients.
The Lagrange method. Exercises page 148
Problem number: 669.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_y]]
\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+2 x y^{\prime }=\frac {1}{x^{2}+1}} \] With initial conditions \begin {align*} \left [y \left (\infty \right ) = \frac {\pi ^{2}}{8}, y^{\prime }\left (0\right ) = 0\right ] \end {align*}
✓ Solution by Maple
Time used: 0.063 (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)
\[ y \left (x \right ) = \frac {\arctan \left (x \right )^{2}}{2} \]
✓ Solution by Mathematica
Time used: 0.044 (sec). Leaf size: 13
DSolve[{(1+x^2)*y''[x]+2*x*y'[x]==1/(1+x^2),{y[Infinity]==Pi^2/8,y'[0]==0}},y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\arctan (x)^2}{2} \]