23.3.403 problem 407

Internal problem ID [6117]
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 : 407
Date solved : Friday, October 03, 2025 at 01:46:27 AM
CAS classification : [_Jacobi]

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 45

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

\[ y = c_1 \operatorname {hypergeom}\left (\left [k -p , 1+k +p \right ], \left [1+k \right ], x\right )+c_2 \,x^{-k} \operatorname {hypergeom}\left (\left [-p , 1+p \right ], \left [1-k \right ], x\right ) \]

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 41

ode=(-k + p)*(1 + k + p)*y[x] + (1 + k)*(1 - 2*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 (-((x-1) x))^{-k/2} (c_1 P_p^k(2 x-1)+c_2 Q_p^k(2 x-1)) \end{align*}

✗ Sympy

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

False