55.29.3 problem 63

Internal problem ID [13836]
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 : 63
Date solved : Thursday, October 02, 2025 at 08:07:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} b x y+a y^{\prime }+x y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 39

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

\[ y = \left (\operatorname {BesselJ}\left (\frac {a}{2}-\frac {1}{2}, \sqrt {b}\, x \right ) c_1 +\operatorname {BesselY}\left (\frac {a}{2}-\frac {1}{2}, \sqrt {b}\, x \right ) c_2 \right ) x^{-\frac {a}{2}+\frac {1}{2}} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 54

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

\begin{align*} y(x)&\to x^{\frac {1}{2}-\frac {a}{2}} \left (c_1 \operatorname {BesselJ}\left (\frac {a-1}{2},\sqrt {b} x\right )+c_2 \operatorname {BesselY}\left (\frac {a-1}{2},\sqrt {b} x\right )\right ) \end{align*}

✗ Sympy

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

TypeError : invalid input: 1 - a