87.14.24 problem 24

Internal problem ID [23543]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 109
Problem number : 24
Date solved : Thursday, October 02, 2025 at 09:42:49 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*}

Using reduction of order method given that one solution is

\begin{align*} y&=2 x^{2}-1 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 31

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);
 

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

✓ Mathematica. Time used: 0.083 (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]
 

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

✗ 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