23.3.570 problem 578

Internal problem ID [6284]
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 : 578
Date solved : Friday, October 03, 2025 at 01:58:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} -\left (4 k^{2}+\left (-4 p^{2}+1\right ) \left (-x^{2}+1\right )\right ) y-8 x \left (-x^{2}+1\right ) y^{\prime }+4 \left (-x^{2}+1\right )^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 21

ode:=-(4*k^2+(-4*p^2+1)*(-x^2+1))*y(x)-8*x*(-x^2+1)*diff(y(x),x)+4*(-x^2+1)^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 28

ode=-((4*k^2 + (1 - 4*p^2)*(1 - x^2))*y[x]) - 8*x*(1 - x^2)*D[y[x],x] + 4*(1 - x^2)^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 P_{p-\frac {1}{2}}^k(x)+c_2 Q_{p-\frac {1}{2}}^k(x) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
k = symbols("k") 
p = symbols("p") 
y = Function("y") 
ode = Eq(-8*x*(1 - x**2)*Derivative(y(x), x) + 4*(1 - x**2)**2*Derivative(y(x), (x, 2)) + (-4*k**2 - (1 - 4*p**2)*(1 - x**2))*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False