55.28.27 problem 37

Internal problem ID [13810]
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 : 37
Date solved : Friday, October 03, 2025 at 06:54:56 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.198 (sec). Leaf size: 255

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

\[ y = {\mathrm e}^{-\frac {x \left (2 a^{2} x^{2} \operatorname {csgn}\left (a \right )+3 a b x \,\operatorname {csgn}\left (a \right )+2 a^{2} x^{2}+3 a b x -12 \,\operatorname {csgn}\left (a \right ) \alpha \right )}{12 a}} \left (c_2 \,{\mathrm e}^{\frac {\operatorname {csgn}\left (a \right ) x \left (2 a^{2} x^{2}+3 a b x -12 \alpha \right )}{6 a}} \operatorname {HeunT}\left (\frac {3^{{2}/{3}} \left (2 a^{2} \gamma -b a \beta +\alpha \,b^{2}+2 \alpha ^{2}\right )}{2 a^{2} \left (a^{2}\right )^{{1}/{3}}}, \frac {3 \left (a^{2}-\beta a +b \alpha \right ) \operatorname {csgn}\left (a \right )}{a^{2}}, -\frac {3^{{1}/{3}} \left (b^{2}+8 \alpha \right )}{4 \left (a^{2}\right )^{{2}/{3}}}, -\frac {3^{{2}/{3}} a \left (2 a x +b \right )}{6 \left (a^{2}\right )^{{5}/{6}}}\right )+\operatorname {HeunT}\left (\frac {3^{{2}/{3}} \left (2 a^{2} \gamma -b a \beta +\alpha \,b^{2}+2 \alpha ^{2}\right )}{2 a^{2} \left (a^{2}\right )^{{1}/{3}}}, -\frac {3 \left (a^{2}-\beta a +b \alpha \right ) \operatorname {csgn}\left (a \right )}{a^{2}}, -\frac {3^{{1}/{3}} \left (b^{2}+8 \alpha \right )}{4 \left (a^{2}\right )^{{2}/{3}}}, \frac {3^{{2}/{3}} a \left (2 a x +b \right )}{6 \left (a^{2}\right )^{{5}/{6}}}\right ) c_1 \right ) \]

✗ Mathematica

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

Not solved

✗ Sympy

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

False