54.3.353 problem 1370

Internal problem ID [12648]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1370
Date solved : Wednesday, October 01, 2025 at 02:18:56 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 19

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

\[ y = c_1 \sinh \left (a \,\operatorname {arctanh}\left (x \right )\right )+c_2 \cosh \left (a \,\operatorname {arctanh}\left (x \right )\right ) \]

✓ Mathematica. Time used: 2.551 (sec). Leaf size: 87

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

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

✗ Sympy

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

False