23.3.404 problem 408

Internal problem ID [6118]
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 : 408
Date solved : Friday, October 03, 2025 at 01:46:29 AM
CAS classification : [_Jacobi]

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 51

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

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

✓ Mathematica. Time used: 0.106 (sec). Leaf size: 56

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

\begin{align*} y(x)&\to c_1 \operatorname {Hypergeometric2F1}(-n,a+n,c,x)-(-1)^{-c} c_2 x^{1-c} \operatorname {Hypergeometric2F1}(-c-n+1,a-c+n+1,2-c,x) \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None