54.3.352 problem 1369

Internal problem ID [12647]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1369
Date solved : Wednesday, October 01, 2025 at 02:18:56 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 55

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

\[ y = \sqrt {x^{2}-1}\, \left (\left (\frac {x -1}{x +1}\right )^{-\frac {\sqrt {-a +1}}{2}} c_2 +\left (\frac {x -1}{x +1}\right )^{\frac {\sqrt {-a +1}}{2}} c_1 \right ) \]

✓ Mathematica. Time used: 0.056 (sec). Leaf size: 82

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

\begin{align*} 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 ) \end{align*}

✓ Sympy. Time used: 0.238 (sec). Leaf size: 100

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*y(x)/(x**2 - 1)**2 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\sqrt {x^{2} - 1} \left (C_{1} \sqrt {x^{2}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2} - \frac {\sqrt {1 - a}}{2}, 1 - \frac {\sqrt {1 - a}}{2} \\ \frac {3}{2} \end {matrix}\middle | {x^{2}} \right )} + C_{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {\sqrt {1 - a}}{2}, \frac {1}{2} - \frac {\sqrt {1 - a}}{2} \\ \frac {1}{2} \end {matrix}\middle | {x^{2}} \right )}\right ) \sqrt [4]{x^{2}} e^{- \frac {\sqrt {1 - a} \log {\left (x^{2} - 1 \right )}}{2}}}{\sqrt {x}} \]