23.3.430 problem 435

Internal problem ID [6144]
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 : 435
Date solved : Friday, October 03, 2025 at 01:46:38 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 49

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

\[ y = \left (\operatorname {LegendreQ}\left (\frac {\sqrt {16 a^{2}+1}}{4}-\frac {1}{2}, \frac {3}{4}, x\right ) c_2 +\operatorname {LegendreP}\left (\frac {\sqrt {16 a^{2}+1}}{4}-\frac {1}{2}, \frac {3}{4}, x\right ) c_1 \right ) \left (x^{2}-1\right )^{{3}/{8}} \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 66

ode=2*a^2*y[x] - x*D[y[x],x] + 2*(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 )^{3/8} \left (c_1 P_{\frac {1}{4} \left (\sqrt {16 a^2+1}-2\right )}^{\frac {3}{4}}(x)+c_2 Q_{\frac {1}{4} \left (\sqrt {16 a^2+1}-2\right )}^{\frac {3}{4}}(x)\right ) \end{align*}

✗ Sympy

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

False