61.29.33 problem 142

Internal problem ID [12563]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 142
Date solved : Thursday, March 13, 2025 at 11:43:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{3}+B \,x^{2}+C x +d \right ) y&=0 \end{align*}

✓ Maple. Time used: 1.159 (sec). Leaf size: 232

ode:=x^2*diff(diff(y(x),x),x)+x*(a*x^2+b*x+c)*diff(y(x),x)+(A*x^3+B*x^2+C*x+d)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{-\frac {c}{2}+\frac {1}{2}} {\mathrm e}^{\frac {x \left (-a^{2} x -2 a b +2 A \right )}{2 a}} \left (c_{1} x^{\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}} \operatorname {HeunB}\left (\sqrt {c^{2}-2 c -4 d +1}, \frac {\sqrt {2}\, \left (-a b +2 A \right )}{a^{{3}/{2}}}, -c -\frac {2 A b}{a^{2}}+\frac {2 B}{a}-1+\frac {2 A^{2}}{a^{3}}, \frac {\sqrt {2}\, \left (-b c +2 C \right )}{\sqrt {a}}, -\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right )+c_{2} x^{-\frac {\sqrt {c^{2}-2 c -4 d +1}}{2}} \operatorname {HeunB}\left (-\sqrt {c^{2}-2 c -4 d +1}, \frac {\sqrt {2}\, \left (-a b +2 A \right )}{a^{{3}/{2}}}, -c -\frac {2 A b}{a^{2}}+\frac {2 B}{a}-1+\frac {2 A^{2}}{a^{3}}, \frac {\sqrt {2}\, \left (-b c +2 C \right )}{\sqrt {a}}, -\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right )\right ) \]

✗ Mathematica

ode=x^2*D[y[x],{x,2}]+x*(a*x^2+b*x+c)*D[y[x],x]+(A*x^3+B*x^2+C0*x+d)*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") 
C = symbols("C") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
d = symbols("d") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(a*x**2 + b*x + c)*Derivative(y(x), x) + (A*x**3 + B*x**2 + C*x + d)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None