83.41.8 problem 2 (vii)

Internal problem ID [19417]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VIII. Linear equations of second order. Excercise at end of chapter VIII. Page 141
Problem number : 2 (vii)
Date solved : Monday, March 31, 2025 at 07:13:06 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 19

ode:=(-x^2+1)*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+a^2*y(x)/(-x^2+1) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 2.043 (sec). Leaf size: 50

ode=(1-x^2)*D[y[x],{x,2}]-2*x*D[y[x],x]+a^2*y[x]/(1-x^2)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 \cos \left (\frac {1}{2} a (\log (1-x)-\log (x+1))\right )+c_2 \sin \left (\frac {1}{2} a (\log (1-x)-\log (x+1))\right ) \]

✗ Sympy

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

False