60.3.363 problem 1369

Internal problem ID [11373]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1369
Date solved : Monday, January 27, 2025 at 11:18:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 55

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

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

✓ Solution by Mathematica

Time used: 0.086 (sec). Leaf size: 82

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

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