61.27.24 problem 34

Internal problem ID [12455]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 34
Date solved : Monday, March 31, 2025 at 05:35:06 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.145 (sec). Leaf size: 120

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

\[ y = {\mathrm e}^{-\frac {x \left (\left (a \,x^{2}+3 b -6 c \right ) \operatorname {csgn}\left (a \right )+a \,x^{2}+3 b \right )}{6}} \left (c_2 \operatorname {HeunT}\left (0, 3 \,\operatorname {csgn}\left (a \right ), \frac {a \left (b -2 c \right ) 3^{{1}/{3}}}{\left (a^{2}\right )^{{2}/{3}}}, -\frac {3^{{2}/{3}} \left (a^{2}\right )^{{1}/{6}} x}{3}\right ) {\mathrm e}^{\frac {\operatorname {csgn}\left (a \right ) x \left (a \,x^{2}+3 b -6 c \right )}{3}}+\operatorname {HeunT}\left (0, -3 \,\operatorname {csgn}\left (a \right ), \frac {a \left (b -2 c \right ) 3^{{1}/{3}}}{\left (a^{2}\right )^{{2}/{3}}}, \frac {3^{{2}/{3}} \left (a^{2}\right )^{{1}/{6}} x}{3}\right ) c_1 \right ) \]

✓ Mathematica. Time used: 1.118 (sec). Leaf size: 46

ode=D[y[x],{x,2}]+(a*x^2+b)*D[y[x],x]+c*(a*x^2+b-c)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-c x} \left (c_2 \int _1^xe^{-\frac {1}{3} K[1] \left (a K[1]^2+3 b-6 c\right )}dK[1]+c_1\right ) \]

✗ Sympy

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

False