55.29.22 problem 82

Internal problem ID [13855]
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 : 82
Date solved : Thursday, October 02, 2025 at 08:07:51 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 46

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

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

✓ Mathematica. Time used: 1.059 (sec). Leaf size: 63

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + (2*a*x + b)*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