23.3.378 problem 382

Internal problem ID [6092]
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 : 382
Date solved : Friday, October 03, 2025 at 01:46:19 AM
CAS classification : [_Gegenbauer]

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 69

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

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

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 86

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

\begin{align*} y(x)&\to \left (x^2-1\right )^{\frac {a+2}{4}} \left (c_1 P_{\frac {1}{2} \left (\sqrt {a^2+2 a+4 b+1}-1\right )}^{\frac {a+2}{2}}(x)+c_2 Q_{\frac {1}{2} \left (\sqrt {a^2+2 a+4 b+1}-1\right )}^{\frac {a+2}{2}}(x)\right ) \end{align*}

✗ Sympy

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

False