3.368 problem 1369

Internal problem ID [8948]

Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1369.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]

Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {a y}{\left (x^{2}-1\right )^{2}}=0} \end {gather*}

Solution by Maple

Time used: 0.01 (sec). Leaf size: 61

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

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

Solution by Mathematica

Time used: 0.05 (sec). Leaf size: 82

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

\begin{align*} y(x)\to \frac {1}{2} \sqrt {1-x^2} \left (\frac {c_2 (x+1)^{\sqrt {1-a}} (1-x)^{-\sqrt {1-a}}}{\sqrt {1-a}}+2 c_1\right ) e^{-\sqrt {1-a} \tanh ^{-1}(x)} \\ \end{align*}