23.3.379 problem 383

Internal problem ID [6093]
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 : 383
Date solved : Friday, October 03, 2025 at 01:46:20 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (\operatorname {c0} \,x^{2}+\operatorname {b0} x +\operatorname {a0} \right ) y+a x y^{\prime }+\left (-x^{2}+1\right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.123 (sec). Leaf size: 132

ode:=(c0*x^2+b0*x+a0)*y(x)+a*x*diff(y(x),x)+(-x^2+1)*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{\sqrt {\operatorname {c0}}\, x} \left (c_2 \left (\frac {x}{2}-\frac {1}{2}\right )^{1+\frac {a}{4}} \left (\frac {x}{2}+\frac {1}{2}\right )^{-\frac {a}{4}} \left (x^{2}-1\right )^{\frac {a}{4}} \operatorname {HeunC}\left (4 \sqrt {\operatorname {c0}}, -\frac {a}{2}-1, 1+\frac {a}{2}, -2 \operatorname {b0} , -\operatorname {a0} +\operatorname {b0} -\frac {a^{2}}{8}-\operatorname {c0} +\frac {1}{2}, \frac {x}{2}+\frac {1}{2}\right )+c_1 \left (x^{2}-1\right )^{1+\frac {a}{2}} \operatorname {HeunC}\left (4 \sqrt {\operatorname {c0}}, 1+\frac {a}{2}, 1+\frac {a}{2}, -2 \operatorname {b0} , -\operatorname {a0} +\operatorname {b0} -\frac {a^{2}}{8}-\operatorname {c0} +\frac {1}{2}, \frac {x}{2}+\frac {1}{2}\right )\right ) \]

✓ Mathematica. Time used: 0.354 (sec). Leaf size: 185

ode=(a0 + b0*x + c0*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]
 

\begin{align*} y(x)&\to \frac {1}{2} (x-1)^{-a/4} e^{\sqrt {\text {c0}} x} \left (c_2 \left (x^2-1\right )^{a/4} (x+1)^{\frac {a}{4}+1} \text {HeunC}\left [\frac {a^2}{4}+a \left (\sqrt {\text {c0}}+\frac {1}{2}\right )+\text {a0}-\text {b0}+\text {c0}+4 \sqrt {\text {c0}},4 \sqrt {\text {c0}}-2 \text {b0},\frac {a+4}{2},-\frac {a}{2},4 \sqrt {\text {c0}},\frac {x+1}{2}\right ]+2 c_1 (x-1)^{a/4} \text {HeunC}\left [a \left (-\sqrt {\text {c0}}\right )+\text {a0}-\text {b0}+\text {c0},-2 \left (a \sqrt {\text {c0}}+\text {b0}\right ),-\frac {a}{2},-\frac {a}{2},4 \sqrt {\text {c0}},\frac {x+1}{2}\right ]\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
a0 = symbols("a0") 
b0 = symbols("b0") 
c0 = symbols("c0") 
y = Function("y") 
ode = Eq(a*x*Derivative(y(x), x) + (1 - x**2)*Derivative(y(x), (x, 2)) + (a0 + b0*x + c0*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False