23.3.376 problem 380

Internal problem ID [6090]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 380
Date solved : Tuesday, September 30, 2025 at 02:21:21 PM
CAS classification : [_Gegenbauer]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 27

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

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

✓ Mathematica. Time used: 0.49 (sec). Leaf size: 99

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

\begin{align*} y(x)&\to \frac {\left (1-x^2\right )^{\frac {1}{2}-\frac {1}{2} \sqrt {(a-1)^2}} \left (x^2-1\right )^{-a/2} \left (2 \sqrt {(a-1)^2} c_1 (1-x)^{\sqrt {(a-1)^2}}+c_2 (x+1)^{\sqrt {(a-1)^2}}\right )}{2 \sqrt {(a-1)^2}} \end{align*}

✗ Sympy

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

False