23.3.372 problem 376

Internal problem ID [6086]
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 : 376
Date solved : Friday, October 03, 2025 at 01:46:13 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 61

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

\[ y = c_1 \operatorname {hypergeom}\left (\left [-n , 1+a +b +n \right ], \left [b +1\right ], \frac {x}{2}+\frac {1}{2}\right )+c_2 \left (\frac {x}{2}+\frac {1}{2}\right )^{-b} \operatorname {hypergeom}\left (\left [-n -b , 1+a +n \right ], \left [1-b \right ], \frac {x}{2}+\frac {1}{2}\right ) \]

✓ Mathematica. Time used: 0.125 (sec). Leaf size: 69

ode=n*(1 + a + b + n)*y[x] + (-a + b - (2 + a + b)*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 2^a c_2 (x-1)^{-a} \operatorname {Hypergeometric2F1}\left (-a-n,b+n+1,1-a,\frac {1-x}{2}\right )+c_1 \operatorname {Hypergeometric2F1}\left (-n,a+b+n+1,a+1,\frac {1-x}{2}\right ) \end{align*}

✗ Sympy

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

False