23.3.436 problem 441

Internal problem ID [6150]
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 : 441
Date solved : Saturday, October 04, 2025 at 04:29:04 PM
CAS classification : [_Jacobi]

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 66

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

\[ y = c_1 \operatorname {hypergeom}\left (\left [-1-\frac {\sqrt {2 a +4}}{2}, -1+\frac {\sqrt {2 a +4}}{2}\right ], \left [-\frac {1}{2}\right ], x\right )+c_2 \,x^{{3}/{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {2 a +4}}{2}, \frac {1}{2}+\frac {\sqrt {2 a +4}}{2}\right ], \left [\frac {5}{2}\right ], x\right ) \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 75

ode=a*y[x] - (1 - 2*x)*D[y[x],x] + 2*(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))^{3/4} \left (c_1 P_{\frac {\sqrt {a+2}}{\sqrt {2}}-\frac {1}{2}}^{\frac {3}{2}}(2 x-1)+c_2 Q_{\frac {\sqrt {a+2}}{\sqrt {2}}-\frac {1}{2}}^{\frac {3}{2}}(2 x-1)\right ) \end{align*}

✗ Sympy

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

False