60.3.232 problem 1248

Internal problem ID [11228]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1248
Date solved : Sunday, March 30, 2025 at 07:57:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.149 (sec). Leaf size: 123

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

\[ y = {\mathrm e}^{\sqrt {-b}\, x} \left (\left (x^{2}-1\right )^{-\frac {a}{4}} \left (\frac {x}{2}+\frac {1}{2}\right )^{-\frac {a}{4}+1} \operatorname {HeunC}\left (4 \sqrt {-b}, 1-\frac {a}{2}, \frac {a}{2}-1, 2 c , d -c -\frac {a^{2}}{8}+b +\frac {1}{2}, \frac {x}{2}+\frac {1}{2}\right ) \left (\frac {x}{2}-\frac {1}{2}\right )^{\frac {a}{4}} c_2 +\operatorname {HeunC}\left (4 \sqrt {-b}, \frac {a}{2}-1, \frac {a}{2}-1, 2 c , d -c -\frac {a^{2}}{8}+b +\frac {1}{2}, \frac {x}{2}+\frac {1}{2}\right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.542 (sec). Leaf size: 192

ode=(d + c*x + b*x^2)*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]
 

\[ y(x)\to \frac {1}{2} e^{\sqrt {-b} x} \left (c_2 (x-1)^{a/4} \left (x^2-1\right )^{-a/4} (x+1)^{1-\frac {a}{4}} \text {HeunC}\left [\frac {1}{4} a \left (a-4 \sqrt {-b}-2\right )-b+4 \sqrt {-b}+c-d,2 \left (2 \sqrt {-b}+c\right ),2-\frac {a}{2},\frac {a}{2},4 \sqrt {-b},\frac {x+1}{2}\right ]+2 c_1 \text {HeunC}\left [a \sqrt {-b}-b+c-d,2 \left (a \sqrt {-b}+c\right ),\frac {a}{2},\frac {a}{2},4 \sqrt {-b},\frac {x+1}{2}\right ]\right ) \]

✗ Sympy

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

False