23.3.556 problem 563

Internal problem ID [6270]
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 : 563
Date solved : Friday, October 03, 2025 at 01:58:07 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (\operatorname {a4} \,x^{4}+\operatorname {a2} \,x^{2}+\operatorname {a0} \right ) y-2 x \left (a^{2}-x^{2}\right ) y^{\prime }+\left (a^{2}-x^{2}\right )^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.128 (sec). Leaf size: 144

ode:=(a4*x^4+a2*x^2+a0)*y(x)-2*x*(a^2-x^2)*diff(y(x),x)+(a^2-x^2)^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\operatorname {HeunC}\left (0, \frac {1}{2}, \frac {\sqrt {-a^{4} \operatorname {a4} -a^{2} \operatorname {a2} -\operatorname {a0}}}{a}, \frac {a^{2} \operatorname {a4}}{4}, \frac {a^{2}-\operatorname {a0}}{4 a^{2}}, \frac {x^{2}}{a^{2}}\right ) c_2 x +\operatorname {HeunC}\left (0, -\frac {1}{2}, \frac {\sqrt {-a^{4} \operatorname {a4} -a^{2} \operatorname {a2} -\operatorname {a0}}}{a}, \frac {a^{2} \operatorname {a4}}{4}, \frac {a^{2}-\operatorname {a0}}{4 a^{2}}, \frac {x^{2}}{a^{2}}\right ) c_1 \right ) \left (a^{2}-x^{2}\right )^{\frac {\sqrt {-a^{4} \operatorname {a4} -a^{2} \operatorname {a2} -\operatorname {a0}}}{2 a}} \]

✗ Mathematica

ode=(a0 + a2*x^2 + a4*x^4)*y[x] - 2*x*(a^2 - x^2)*D[y[x],x] + (a^2 - x^2)^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
a0 = symbols("a0") 
a2 = symbols("a2") 
a4 = symbols("a4") 
y = Function("y") 
ode = Eq(-2*x*(a**2 - x**2)*Derivative(y(x), x) + (a**2 - x**2)**2*Derivative(y(x), (x, 2)) + (a0 + a2*x**2 + a4*x**4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False