49.15.11 problem 7

Internal problem ID [7697]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 3. Linear equations with variable coefficients. Page 130
Problem number : 7
Date solved : Wednesday, March 05, 2025 at 04:50:41 AM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 33

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

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

✓ Mathematica. Time used: 0.051 (sec). Leaf size: 45

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

\[ y(x)\to c_1 \cosh \left (\alpha \log \left (\sqrt {x^2-1}+x\right )\right )+i c_2 \sinh \left (\alpha \log \left (\sqrt {x^2-1}+x\right )\right ) \]

✗ Sympy

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

False