55.29.25 problem 85

Internal problem ID [13858]
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 : 85
Date solved : Friday, October 03, 2025 at 06:55:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.027 (sec). Leaf size: 143

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

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

✓ Mathematica. Time used: 0.054 (sec). Leaf size: 134

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

\begin{align*} y(x)&\to \frac {e^{-\frac {1}{2} x \left (-\frac {2 c}{a}+a x+2 b\right )} \left (c_2 \operatorname {Hypergeometric1F1}\left (-\frac {-2 a^3+d a^2-b c a+c^2}{2 a^3},\frac {1}{2},\frac {\left (x a^2+b a-2 c\right )^2}{2 a^3}\right )+c_1 \operatorname {HermiteH}\left (\frac {-2 a^3+d a^2-b c a+c^2}{a^3},\frac {x a^2+b a-2 c}{\sqrt {2} a^{3/2}}\right )\right )}{x} \end{align*}

✗ Sympy

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

False