23.3.464 problem 470

Internal problem ID [6178]
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 : 470
Date solved : Friday, October 03, 2025 at 01:48:09 AM
CAS classification : [_Jacobi]

\begin{align*} -\left (k -p \right ) \left (1+k +p \right ) y+2 \left (1-\left (3-2 k \right ) x \right ) y^{\prime }+4 \left (1-x \right ) x y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 65

ode:=-(k-p)*(1+k+p)*y(x)+2*(1-(3-2*k)*x)*diff(y(x),x)+4*(1-x)*x*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (-1+x \right )^{\frac {k}{2}} \left (\operatorname {LegendreQ}\left (\frac {\sqrt {4 p^{2}-8 k +4 p +1}}{2}-\frac {1}{2}, k , \sqrt {x}\right ) c_2 +\operatorname {LegendreP}\left (\frac {\sqrt {4 p^{2}-8 k +4 p +1}}{2}-\frac {1}{2}, k , \sqrt {x}\right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.108 (sec). Leaf size: 130

ode=-((k - p)*(1 + k + p)*y[x]) + 2*(1 - (3 - 2*k)*x)*D[y[x],x] + 4*(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}\left (\frac {1}{4} \left (-2 k-\sqrt {(2 p+1)^2-8 k}+1\right ),\frac {1}{4} \left (-2 k+\sqrt {(2 p+1)^2-8 k}+1\right ),\frac {1}{2},x\right )+i c_2 \sqrt {x} \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (-2 k-\sqrt {(2 p+1)^2-8 k}+3\right ),\frac {1}{4} \left (-2 k+\sqrt {(2 p+1)^2-8 k}+3\right ),\frac {3}{2},x\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
k = symbols("k") 
p = symbols("p") 
y = Function("y") 
ode = Eq(x*(4 - 4*x)*Derivative(y(x), (x, 2)) + (-k + p)*(k + p + 1)*y(x) + (-2*x*(3 - 2*k) + 2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False