62.31.2 problem Ex 2

Internal problem ID [12895]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VIII, Linear differential equations of the second order. Article 54. Change of independent variable. Page 127
Problem number : Ex 2
Date solved : Wednesday, March 05, 2025 at 08:50:47 PM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+4 y&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 50

ode:=(-x^2+1)*diff(diff(y(x),x),x)-x*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (8 x^{3}-4 x \right ) c_{2} \sqrt {x^{2}-1}+\left (8 x^{4}-8 x^{2}+1\right ) c_{2} +c_{1}}{\left (x +\sqrt {x^{2}-1}\right )^{2}} \]

✓ Mathematica. Time used: 0.099 (sec). Leaf size: 65

ode=(1-x^2)*D[y[x],{x,2}]-x*D[y[x],x]+4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x) + (1 - x**2)*Derivative(y(x), (x, 2)) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False