60.3.359 problem 1365

Internal problem ID [11369]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1365
Date solved : Monday, January 27, 2025 at 11:18:16 PM
CAS classification : [_Halm]

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 55

dsolve(diff(diff(y(x),x),x) = -a/(x^2+1)^2*y(x),y(x), singsol=all)
 

\[ y = \left (\left (\frac {x +i}{-x +i}\right )^{\frac {\sqrt {a +1}}{2}} c_{1} +\left (\frac {x +i}{-x +i}\right )^{-\frac {\sqrt {a +1}}{2}} c_{2} \right ) \sqrt {x^{2}+1} \]

✓ Solution by Mathematica

Time used: 0.088 (sec). Leaf size: 86

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

\[ y(x)\to \exp \left (\int _1^x\frac {K[1]+i \sqrt {a+1}}{K[1]^2+1}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {K[1]+i \sqrt {a+1}}{K[1]^2+1}dK[1]\right )dK[2]+c_1\right ) \]