23.3.465 problem 471

Internal problem ID [6179]
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 : 471
Date solved : Friday, October 03, 2025 at 01:48:10 AM
CAS classification : [_Jacobi]

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

✓ Maple. Time used: 0.047 (sec). Leaf size: 66

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

\[ y = \left (\sqrt {x}\, \operatorname {HeunC}\left (0, \frac {1}{2}, \frac {3}{2}+\frac {a}{2}, -\frac {k^{2}}{4}, \frac {a}{8}-\frac {b}{4}+\frac {5}{8}, x\right ) c_2 +\operatorname {HeunC}\left (0, -\frac {1}{2}, \frac {3}{2}+\frac {a}{2}, -\frac {k^{2}}{4}, \frac {a}{8}-\frac {b}{4}+\frac {5}{8}, x\right ) c_1 \right ) \left (-1+x \right )^{\frac {3}{2}+\frac {a}{2}} \]

✗ Mathematica

ode=(b + k^2*x)*y[x] + 2*(1 + a*x)*D[y[x],x] + 4*(1 - x)*x*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") 
b = symbols("b") 
k = symbols("k") 
y = Function("y") 
ode = Eq(x*(4 - 4*x)*Derivative(y(x), (x, 2)) + (b + k**2*x)*y(x) + (2*a*x + 2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False