23.3.396 problem 400

Internal problem ID [6110]
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 : 400
Date solved : Friday, October 03, 2025 at 01:46:25 AM
CAS classification : [_Jacobi]

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 51

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

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

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 26

ode=p*(1 + p)*y[x] + (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 c_1 \operatorname {LegendreP}(p,2 x-1)+c_2 \operatorname {LegendreQ}(p,2 x-1) \end{align*}

✗ Sympy

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

False