61.28.24 problem 84

Internal problem ID [12505]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 84
Date solved : Thursday, March 13, 2025 at 11:43:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{2}+B x +\operatorname {C0} \right ) y&=0 \end{align*}

✓ Maple. Time used: 1.099 (sec). Leaf size: 186

ode:=x*diff(diff(y(x),x),x)+(a*x^2+b*x+c)*diff(y(x),x)+(A*x^2+B*x+C0)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{\frac {x \left (-a^{2} x -2 a b +2 A \right )}{2 a}} \left (\operatorname {HeunB}\left (-c +1, -\frac {\sqrt {2}\, \left (-a b +2 A \right )}{a^{{3}/{2}}}, \frac {\left (-c -1\right ) a^{3}+2 B \,a^{2}-2 A a b +2 A^{2}}{a^{3}}, \frac {\left (b c -2 \operatorname {C0} \right ) \sqrt {2}}{\sqrt {a}}, \frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right ) x^{-c +1} c_{2} +\operatorname {HeunB}\left (c -1, -\frac {\sqrt {2}\, \left (-a b +2 A \right )}{a^{{3}/{2}}}, \frac {\left (-c -1\right ) a^{3}+2 B \,a^{2}-2 A a b +2 A^{2}}{a^{3}}, \frac {\left (b c -2 \operatorname {C0} \right ) \sqrt {2}}{\sqrt {a}}, \frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right ) c_{1} \right ) \]

✗ Mathematica

ode=x*D[y[x],{x,2}]+(a*x^2+b*x+c)*D[y[x],x]+(A*x^2+B*x+C0)*y[x]==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") 
C0 = symbols("C0") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + (A*x**2 + B*x + C0)*y(x) + (a*x**2 + b*x + c)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None